Car will not run with headlights on but will run

Have you had vehicle check for DTC'S diagnostic trouble codes . Check fuse's an relays , find wiring diagrams to see what controls these systems ?
Radio/Battery fuse , fuse G 10 amp located in the right instrument panel fuse box . Do you know where this ?
Fuse Block - Right IP - On the right side of the instrument panel
An there is one on the other end of the instrument panel ,driver side .
Fuse Block - Left IP- On the left side of the instrument panel The fuel an temp . gauges not working is probably the step motors inside instrument cluster . GM had problem with this 03 to 06 .
How to repair GM Instrument Cluster Speedometer Gauge Stepper

Daytime Running Lamps (DRL)
The ambient light sensor is a light sensitive transistor that varies its voltage signal to the body control module (BCM) in response to changes to the outside (ambient) light level. When the BCM receives this signal it will either turn on the daytime running lights (DRL) or the headlights. Any function or condition that turns on the headlights will cancel the daytime running lamps operation. With the headlight switch in the OFF position, the headlights will either be turned ON or OFF, after an approximate 30 second delay depending on whether daylight or low light conditions are sensed. The HTR A/C IGN fuse in the underhood junction block supplies ignition positive voltage to the daytime running lamp (DRL) relay coil circuit. The DRL relay switch contacts are connected to the low beam circuit. When the headlamp switch is placed in the OFF position, the BCM will either turn on or off the daytime running lamps, depending on whether daylight or low light conditions are sensed. The BCM energizes the DRL relay, closing the switch contacts and grounding the low beam circuit. Ground is provided at G101. The DRL illuminate the low beam headlights at a reduced intensity. The DRL will operate when the ignition switch is in the ON position, the gear selector is not in the PARK position and the parking brake is released. When these conditions have been met and the ambient light sensor indicates daytime conditions, the DRL will illuminate.
DRL operation in Manual Transmission equipped vehicles will occur when the ignition switch is in the ON position, and the parking brake is released. - check fuse's an relay first.
Door Lock System Components
The power door lock system consists of the following components:
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Driver door lock switch

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Front Passenger door lock switch

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Body Control Module (BCM)

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Relays: Driver Door Unlock--if equipped, Door Unlock, and Door Lock

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Door Lock fuse 15A

That piece you think is a ceramic DRL module is a resistor , When in DRL mode voltage to the low beam headlamps circuit goes through that resistor dimming the headlamps from low beam to daytime running lamps . It sounds to me like you need a new resistor ,but without doing some electrical testing it could be a lot of things . The DRL relay in the under hood fuse box could have a problem , could be the BCM - body control module , the ambient light sensor etc... Here is how daytime running lamps work .
Daytime Running Lamps (DRL)
The ambient light sensor is a light sensitive transistor that varies its voltage signal to the body control module (BCM) in response to changes to the outside (ambient) light level. When the BCM receives this signal it will either turn on the daytime running lights (DRL) or the headlights. Any function or condition that turns on the headlights will cancel the daytime running lamps operation. With the headlight switch in the OFF position, the headlights will either be turned ON or OFF, after an approximate 30 second delay depending on whether daylight or low light conditions are sensed. The HTR A/C IGN fuse in the underhood junction block supplies ignition positive voltage to the daytime running lamp (DRL) relay coil circuit. The DRL relay switch contacts are connected to the low beam circuit. When the headlamp switch is placed in the OFF position, the BCM will either turn on or off the daytime running lamps, depending on whether daylight or low light conditions are sensed. The BCM energizes the DRL relay, closing the switch contacts and grounding the low beam circuit. Ground is provided at G101. The DRL illuminate the low beam headlights at a reduced intensity. The DRL will operate when the ignition switch is in the ON position, the gear selector is not in the PARK position and the parking brake is released. When these conditions have been met and the ambient light sensor indicates daytime conditions, the DRL will illuminate.

Things that could be causing this
1. You have DRL main relay damaged
2. You are leaving the lights in the parking light position, not the off position.

Hi,
Likely the Headlights work through the DRL switch, too. Replace the switch (get a used one from a used parts/salvage/wrecking yard) or from a dealer...might call an auto parts store, too.

GL,
Td

Headlamps
The headlamps are operated when the driver places the headlamp switch in the HEADLAMP position. The LT HDLP and RT HDLP fuse in the IP fuse block, supply battery positive voltage to both the left and right headlamps. The low beam and high beam circuits continue to the headlamp switch. Depending on the position of the dimmer switch, either the low or high beam circuit will now have a path to ground. Ground is provided at G204. The low beam and high beam circuits also provide the body control module (BCM) with both low and high beam inputs. The headlamp switch includes the dimmer and the flash-to-pass switch.
If the headlight switch is left in the HEADLAMP position, the headlights will remain on until turned off or the battery runs dead.
Daytime Running Lamps (DRL)
Any function that turns on the headlights will cancel the daytime running lamps operation. The LT HDLP fuse in the IP fuse block supplies battery positive voltage to the daytime running lamp (DRL) relay coil circuit. The DRL relay switch contacts are connected to the low beam circuit. When the headlamp switch is placed in the OFF position, the BCM will turn on the daytime running lamps. The BCM energizes the DRL relay, closing the switch contacts and grounding the low beam circuit at G204. The DRL illuminate the low beam headlights at a reduced intensity. The DRL will operate when the ignition switch is in the RUN position, the gear selector is not in the PARK position and the parking brake is released. When these conditions have been met the DRL will illuminate.
DRL operation in Manual Transmission equipped vehicles will occur when the ignition switch is in the RUN position, and the parking brake is released.

Probably the dimmer switch . But without testing that would be just a guess . Turnning the headlamps on Any function that turns on the headlights will cancel the daytime running lamps operation..
No ,don't think the DRL relay has anything to do with it .

View wiring diagrams an testing the electrical circuits at the dimmer switch would be the proper way to diagnose .

[Information] For a 2002 Toyota Camry, the Daytime Running Lights (DRL) operate utilizing the High Beams (inner Lamps closer to the center of the vehicle front) at a reduced power.

[The troubleshooting process:] Review the Owner's Manual and ensure the test configuration matches the following criteria:
1. According to the 2002 Toyota Camry Owner's Manual (Check your Owners Manual):
"The DRL system will make your headlights come on at a reduced brightness when:
The light switch s in the "DRL" position.
The ignition is on with the engine running.
The headlight switch is off.
The parking brake is released."

2. Set the Test Conditions according to Step 1 above.

3. Check the headlight bulbs to ensure they are not burned out. If one burns out, there is a fair chance that the other would have burned out at the same time. Replace as necessary. In this case, if the high beams operate, the DRLs should also operate.
4. Test the system, ensuring the step 1 conditions are set. (Engine running, Multi-function Switch on Turn Signals in "DRL" position, headlight switch off (repeating what the manual says -- it should be obvious that in the "DRL" position, the headlight switch should be off / do not have the switch positioned to "High Beam"), and the Parking Brake "off").
5. If DRL lights do not work, check the engine compartment Fuse and Relay Box. there is a 5 Amp DRL fuse -- replace it with the spare. Test (step 4).
6. If it does not work, swap around or replace relays -- there are three (3). Any one of them could be bad (stuck or burned out). Test (step 4).
7. Other less likely possible issues could be that the Multi-function Switch is broken or a connection / wire is broken. [Beyond the scope of this process.]

My problem was that the test configuration did not have the engine running and parking brake off - Step 1. Problem solved with much thought, repetitive tests on the vehicle and going back to the basics -- what does the Owner's Manual say and where are the applicable fuses and relays located?

[Information] For a 2002 Toyota Camry, the Daytime Running Lights (DRL) operate utilizing the High Beams (inner Lamps closer to the center of the vehicle front) at a reduced power.

[The troubleshooting process:] Review the Owner's Manual and ensure the test configuration matches the following criteria:
1. According to the 2002 Toyota Camry Owner's Manual (Check your Owners Manual):
"The DRL system will make your headlights come on at a reduced brightness when:
The light switch s in the "DRL" position.
The ignition is on with the engine running.
The headlight switch is off.
The parking brake is released."

2. Set the Test Conditions according to Step 1 above.

3. Check the headlight bulbs to ensure they are not burned out. If one burns out, there is a fair chance that the other would have burned out at the same time. Replace as necessary. In this case, if the high beams operate, the DRLs should also operate.
4. Test the system, ensuring the step 1 conditions are set. (Engine running, Multi-function Switch on Turn Signals in "DRL" position, headlight switch off (repeating what the manual says -- it should be obvious that in the "DRL" position, the headlight switch should be off / do not have the switch positioned to "High Beam"), and the Parking Brake "off").
5. If DRL lights do not work, check the engine compartment Fuse and Relay Box. there is a 5 Amp DRL fuse -- replace it with the spare. Test (step 4).
6. If it does not work, swap around or replace relays -- there are three (3). Any one of them could be bad (stuck or burned out). Test (step 4).
7. Other less likely possible issues could be that the Multi-function Switch is broken or a connection / wire is broken. [Beyond the scope of this process.]

My problem was that the test configuration did not have the engine running and parking brake off - Step 1. Problem solved with much thought, repetitive tests on the vehicle and going back to the basics -- what does the Owner's Manual say and where are the applicable fuses and relays located?

The daytime running lamps in your 2001 Chevrolet Impala are controlled by the Body Control Module (BCM) Here is a description of how the system works:

The ambient light sensor is a light-sensitive transistor that varies its voltage signal to the Body Control Module (BCM) in response to changes in the outside (ambient) light level. When the BCM receives this signal, the BCM turns ON either the DRL or the headlamp relay for auto headlamp operation. Any function or condition that turns ON the headlamps will cancel the DRL operation.
Thirty seconds after you start the ignition with the headlamp switch set in the OFF position, the DRL will perform one of two actions:

Illuminate the high beam headlights at a reduced intensity in daylight conditions
Illuminate the low beam headlights in low light conditions
The DRL/EXT lamps fuse in the top of the engine fuse block supplies battery positive voltage to both the DRL switch contacts and to the DRL relay coil. When the BCM energizes the DRL relay coil, current flows through the right high beam lamp, to the left hand high beam lamp, to the energized DRL relay switch contacts, to ground G101.
The high beam headlamps are now in series, and split the voltage so that the headlamps illuminate at halt intensity. The DRL operates when the ignition switch is in the RUN position, and the parking brake is not set or the transmission is not in park. When these conditions have been met and the ambient light sensor indicates daytime conditions, the DRL will illuminate.

The day time running lamps are your headlights, the parking lights also conventionally known as the running lights serve no purpose during daylight hours and as such are not necessary in the daytime, and are not illuminated by the DRL module. The DRL (Daytime Running Lights) module illuminates the headlamps in a slightly dimmer state than night time dim headlamps, but does not turn on the conventional running lights. The only way you'll get conventional running lights during the day is to turn them on manually. I just reviewed the wiring schematics for that system, and the DRL relay bypasses the dimmer switch when active and prevents high beam operation, and feeds current directly to the low beam headlights, there is no connection to the parking light circuit from the DRL module wiring.

i have a 2000 saturn sl2 & noticed what i think you mean when i bot my saturn. since parking lights always come on, it's connected to them. my 'jellyfish' light goes off when i turn on regular headlights