Why are my turn signals, daytime running lights and low beam lights not work?

Hello. It seems your Chevrolet has either faulty headlamp relay(s) or its headlamps are connected in series; if one don't light, neither will the other one. I can't explain why the 'CHECK ENGINE' light was still on while your truck's engine was off and key was out of ignition switch. Good luck and keep me informed; if you want to.

The DRL - daytime running lamps relay is a PCB type relay - printed circuit board , soldered on the circuit board inside the under hood fuse box. The dealer would tell you that it is non serviceable an you need to replace the whole fuse / relay box. If you know how to solder on circuit boards replacement relay's can be found on the internet . Or if you know how to test electrical circuits ,have a wiring diagram for that circuit you can by pass with a external relay kit.

Daytime Running Lamps (DRL) and Automatic Lamp Control (ALC)
The low beam headlamps are used for DRL operation at a reduced intensity. The DRLs will operate only while the engine is running and the headlamp switch is in the AUTO position. When the DRLs are ON, no other exterior lamps such as the parking lamps, tail lamps, etc. will be on when the DRL are being used. The instrument panel will not be illuminated either.
DRL operation is determined by the ambient light sensor and controlled by the body control module (BCM). The ambient light sensor is a light-sensitive transistor used to monitor outside lighting conditions. The BCM provides a 5-volt reference signal to the sensor while ground is provided at G202. The sensor will vary this voltage signal between 0.2 and 4.9 volts depending on outside lighting conditions. The BCM monitors the ambient light sensor signal circuit to determine if outside lighting conditions are correct for either DRL or ALC operation. When the BCM determines the conditions are met for DRL operation, it applies ground to the DRL relay control circuit energizing the DRL PCB Relay. With the relay energized, battery voltage is applied through the switch side of the relay, the DRL 1 fuse, a resistor wire to the DRL 2 fuse, both low beam headlamp fuses, and the low beam supply voltage circuits illuminating the headlamps at a reduced intensity. Any function or condition that turns on the headlamps will cancel DRL operation.
When the BCM detects low light conditions, it will turn OFF the daytime running lamps and turn ON the low beam headlamps as described above in Low Beam Headlamp Operation. The BCM will also turn ON the low beam headlamps in daylight conditions when the windshield wipers are turned ON.

Your best bet mite be to have a qualified repair shop check it out ,could DTC'S - diagnostic trouble codes stored in the BCM - body control module . Like this one
DTC B2600
DTC B2600 00: Daytime Running Lamp Control Circuit

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.

No on the fuse . The LH HDLP and RH HDLP fuse in the fuse block - under hood, supply battery positive voltage to both low and high beam lamps at all times. For manual operation, when the headlamp switch is in the ON position, the headlamp switch supplies ground directly to the headlamps and to the body control module (BCM) on the headlamp low or high beam signal circuit, depending on the position of the headlamp dimmer switch. The BCM uses this signal to determine if the conditions are present to illuminate the daytime running lamps or park lamps.
The headlamps may be turned on 2 different ways. First, when the driver places the headlamp switch in the ON position, for manual operation. Second, with the headlamp switch placed in the OFF position activating automatic lamp control (ALC).For ALC operation, the BCM receives a signal from the ambient light sensor indicating a low or high ambient, outside, light level. Under low light conditions the BCM grounds the headlamp relay control circuit energizing the headlamp control relay. The now closed relay supplies ground through the headlamp switch to the headlamps and the BCM on the headlamp low or high beam signal circuit, depending on the position of the headlamp dimmer switch. When the BCM grounds the headlamp relay control circuit it also grounds the park lamp relay control circuit illuminating the park lamps. The headlamp relay is grounded at G203.
Daytime Running Lamps (DRL)
The low beam headlamps operate at reduced intensity when in daytime running lamps (DRL) mode. The ground circuit to the switch side of the DRL relay has a resistor in-line to reduce the intensity of the low beam headlamps for DRL operation. The body control module (BCM) receives a signal from the ambient light sensor indicating a low or high ambient, outside, light level. Under daylight conditions the BCM grounds the DRL relay control circuit energizing the DRL control relay. The now closed relay supplies ground directly to the low beam headlamps and to the BCM on the headlamp low beam signal circuit. The low beam headlamps now illuminate with reduced intensity. The DRL relay is grounded at G203. Any time the headlamps are illuminated the DRL will not illuminate. Manual park lamps do not effect DRL operation.

It's the dimmer switch. Which is called the multifunction switch or turn signal switch. Just because it clicks, doesn't mean the it is working properly. If you click and the high beams come on, then click it again and have no low beams, replace the switch. Daytime running lights are controlled by the Body control module.

Daytime running Light module is overheating / it reduces power to low beam to create daytime running light . Ask yourself do I need this - If no law against it = UNPLUG control module . Your low beams will operate normal by turning on switch - Lever / Turn switch will turn on Highbeams but this switch can be Lose internally and Only cure is replace Signal Light Steering wheel switch . Common issue for all Cars is switch stalk / Lever is worn out by Aggressive use. If your car has Automatic light control - re they turn off after 30 second delay when you exit car - Try putting switch in manual mode means you Turn on - off by switch . But I really feel signal stalk is worn . DRCM is a known issue that can be unplugged . Final cheap solution unplug both high beams an Drive on low beam . $$$$$$$ spent chasing small problem will destroy $$$ needed for Bigger issues .

Have you checked the flashers? Maybe ground wire hitting something

its the light switch (combination switch) needs to be replaced

There could be a fuse for each low beam light either under the hood or under the dash. If not, then prob both low beams are bad.

If you have daytime run lights, the following may be of help.

There is a daytime running light module that controls both low beams and the daytime running lights. The low beam bulbs may be OK. The high beams will work fine as they are not controlled through the module.

It may be possible to disable the daytime run lights by cutting the yellow wire going into the module and then connecting it to ground. Tape off the other end.

This will help. Thanks please keep updated.please do rate the solution positively .thank you for using fixya

The blinking light is telling you that you have a headlight out, if you look the service light should be coming on when the light stops blinking.
Replace the head light and all should be fine.