2006 Crown Vic, PI model. With the low beam headlights on, when I pull back on the handle to flash the high beams, all the headlights come on, both high and low. When I push the handle forward to lock the high beams on, both driver's-side headlights go off. Is it most likely the switch inside the column, or something further downstream?

Question edited to add maker.
Question moved to model category.

Here is one idea and some videos.

https://www.justanswer.com/acura/6uky3-acura-mdx-low-beam-headlights-will-not-on.html

Here is how they are suppose to work .
Headlamps
The headlamps may be turned on two different ways. First, when the driver places the headlamp switch in the HEADLAMP position, for normal operation. Second, with the headlamp switch placed in the OFF position, for automatic lamp control (ALC). During ALC, the headlamps will be in day time running lamp (DRL) operation in daylight conditions, or low beam operation in low light conditions. The LH HDLPand RH HDLP fuses, supply battery positive voltage from the underhood junction block to the left and right headlamps. The circuit continues through both low and high beam lamps, then back to the underhood junction block. The low beam and high beam circuits continue to the headlamp switch. 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 switch and the flash-to-pass switch. When the headlamp switch is placed in the HEADLAMP position a path to ground is provided. Ground is provided at G201. Depending on the position of the high/low beam switch, either the low or high beam circuit will now have power and ground.
The LH HDLP fuse in the underhood junction block, supplies battery positive voltage to the automatic lamp control (ALC) headlamp relay coil circuit. The ALC relay switch circuit is connected to the low beam circuit at the headlamp switch. When the headlamp switch is placed in the OFF position, the BCM will automatically turn on the low beams in low light conditions. The BCM energizes the ALC relay, closing the switch contacts and grounding the low beam circuit. Ground is provided at G201. With the headlights in the low beam position, the high beams may be momentarily turned on or flashed with the dimmer switch handle. The flash-to-pass feature is accomplished by pulling the dimmer switch handle toward the driver. The headlights may be switched to high beam with the opposite movement of the dimmer switch lever. When the headlamp dimmer switch handle is pulled toward the driver, the flash-to-pass switch closes grounding the high beam circuit. Ground is provided at G201. Both high beams will remain on until the driver releases the switch handle.
If the driver places the headlight switch in the HEADLAMP position , the headlights will remain on until turned off or the battery runs dead.

Finding a wiring diagram to use along with a DMM - digital multi-meter (voltmeter ) To test the electrical circuits for the headlamps B+ power & ground side of the circuit is the way to find a fix .
Watching a few basic electrical testing video's on youtube may help you .
Basic Electricity for Service Techs Ohm law Current Flow Opens Shorts
How to Use Multimeter to Troubleshoot Common Problems
The Trainer 18 Electrical Testing Tips
Can take years to learn how to trouble shoot automotive electrical systems properly .

Not a fuse problem , same fuse power's low an high beams . Looking at wiring diagrams an using a DVOM - digital volt ohm meter to test the electrical circuit's is the way to find the problem . The light's staying on for 3 or 4 min's. is a function of the BCM - body control module ! Do you know anything about automotive electrical testing ? Do you know what a wiring diagram is an how to read one . Free wiring diagrams here http://www.bbbind.com/free_tsb.html Enter vehicle info. Year , make , model an engine size . Under system click on lighting ,then under subsystem click on headlamps .Click the search button then the blue link. get yourself a cheap DVOM an test the electrical circuit's , this is how electrical problems are solved .
Electric Testing Techniques You Need to Know
Basic Electricity for Service Techs Ohm law Current Flow Opens Shorts

Plus reading factory service repair info for how it should work would be a good idea .

Headlamps
The headlamps may be turned on two different ways. First, when the driver places the headlamp switch in the HEADLAMP position, for normal operation. Second, with the headlamp switch placed in the OFF position, for automatic lamp control (ALC). During ALC, the headlamps will be in day time running lamp (DRL) operation in daylight conditions, or low beam operation in low light conditions. The LH HDLPand RH HDLP fuses, supply battery positive voltage from the underhood junction block to the left and right headlamps. The circuit continues through both low and high beam lamps, then back to the underhood junction block. The low beam and high beam circuits continue to the headlamp switch. 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 switch and the flash-to-pass switch. When the headlamp switch is placed in the HEADLAMP position a path to ground is provided. Ground is provided at G201. Depending on the position of the high/low beam switch, either the low or high beam circuit will now have power and ground.
The LH HDLP fuse in the underhood junction block, supplies battery positive voltage to the automatic lamp control (ALC) headlamp relay coil circuit. The ALC relay switch circuit is connected to the low beam circuit at the headlamp switch. When the headlamp switch is placed in the OFF position, the BCM will automatically turn on the low beams in low light conditions. The BCM energizes the ALC relay, closing the switch contacts and grounding the low beam circuit. Ground is provided at G201. With the headlights in the low beam position, the high beams may be momentarily turned on or flashed with the dimmer switch handle. The flash-to-pass feature is accomplished by pulling the dimmer switch handle toward the driver. The headlights may be switched to high beam with the opposite movement of the dimmer switch lever. When the headlamp dimmer switch handle is pulled toward the driver, the flash-to-pass switch closes grounding the high beam circuit. Ground is provided at G201. Both high beams will remain on until the driver releases the switch handle.
If the driver places the headlight switch in the HEADLAMP position , the headlights will remain on until turned off or the battery runs dead.
Do you know what a factory diagnostic trouble shooting chart is ?

Headlamps Inoperative - Low Beams

Step
Action
Yes
No
Schematic Reference: Headlights/Daytime Running Lights (DRL) Schematics
1
Did you perform the Lighting System Diagnostic System Check?
YES - Go to Step 2
NO - Go to Diagnostic System Check - Lighting Systems
2
Place the headlamp switch in the HEADLAMP position.
Place the headlamp dimmer switch in the LOW position.
Do the low beam headlamps illuminate?
YES - Go to Testing for Intermittent Conditions and Poor Connections in Wiring Systems
NO - Go to Step 3
3
Disconnect the turn/headlamp wiper switch assembly connector C2.
Connect a 15 amp fused jumper between the headlamp low beam signal circuit and a good ground.
Do the low beam headlamps illuminate?
YES - Go to Step 4
NO - Go to Step 5
4
Inspect for poor connections at the harness connector of the turn/headlamp wiper switch assembly. Refer to Testing for Intermittent Conditions and Poor Connections and Connector Repairs in Wiring Systems.
Did you find and correct the condition?
YES - Go to Step 7
NO - Go to Step 6
5
Repair an open or high resistance in the headlamp low beam signal circuit. Refer to Circuit Testing and Wiring Repairs in Wiring Systems.
YES - Did you complete the repair?
NO - Go to Step 7
--
6
Replace the turn/headlamp wiper switch assembly. Refer to Multifunction Turn Signal Lever Replacement - On Vehicle in Steering Wheel and Column.
Did you complete the replacement?
Go to Step 7
--
7
Operate the system in order to verify the repair.
Did you correct the condition?
YES - System OK
NO - Go to Step 3

When either the high or low beam light is out in either side, DRL (daytime running lights) indicator will come on. When one bulb goes out, that's a clue the opposite bulb will burn out soon, so its best to replace both bulbs. If it was a fuse issue, both left and right side would be out. If driver side is OK but passenger side is not, then its a bulb issue. A 2006 Honda civic takes 9006 (HB4) low beam bulbs and 9005 (HB3) high beam bulbs. Cheapest is online from Walmart site-2-store. After procuring, wear latex, vinyl, or ANY gloves to keep your (corrosive) oil skin from coming in contact with headlight glass bulbs. Your skin oil will chemically react and cause shorter life span on glass bulbs. Instruction don't tell you this, so you handle with your hands, then need to replace bulbs 3 years later instead of 5 years (or more) in the future. With gloves on, pop the hood and look behind each headlight. Turn counter clockwise to unlock headlight bulb holders. Its a quarter to half turn to unlock bulb holder lock. Replace one bulb at a time so you don't confuse high beam from low beam. The wire has a tab you need to lift to pull out & disconnect wire connector. You might need a slim narrow-**** (NOT phillips) screwdriver to lift the tab; or use your house key. Snap the wire connector to the new bulb, then twist into headlight housing to secure bulb holders, then move on to the next bulb to repeat the procedure. Pict below, 9005 high beam is on the left, 9006 low beam is on the right. Handle by grasping the plastic base, not the glass bulbs. The raised potrusion on the bottom center is where the tab locks onto wire connector. Replace all 4 bulbs now, but in the future, its likely the low beam will wear out before the high beam. In the future, replace only the pair, not all 4. The factory installs inferior bulbs that go out sooner, hoping you'll take to dealership and they ******** you with higher parts-n-labor fees. As opposed to Walmart online site-2-store, it would cost you AT LEAST triple in parts & labor from dealership to replace all 4 headlight bulbs :-(

Instructions

1. • Turn the headlight switch on and inspect the headlights. Make sure to check both low- and high-beam lights, and then note the position of the bulb in the headlight assembly. Low-beams are in the top cavity of the assembly and high-beams are in the lower cavity.
   • 2Turn off the headlight switch and then remove the keys from the ignition.
     
   • 3
     
   • Open the hood and locate the seam to the headlight assembly and the side fender. Locate the headlight assembly release lever and push in on it slightly to unlock it and then pull it all the way up until it stops.
     
   • 4Place a blanket or pillow on the ground in front of the headlight assembly. Push the assembly outward from behind with one hand while being ready to catch the assembly with the other. The New Beetles use an adhesive sealant to protect the back of the headlights from exposure to moisture, so you'll have to give the assembly a push from behind.
     
   • 5Pull the wire harness plugs off from both the high- and low-beam blades of the respective bulbs.
     
   • 6Place the assembly on the blanket or pillow. Release the wire clip to the faulty headlight and then remove the headlight bulb.
     
   • 7Open the package of the new H1 bulb replacement without touching the glass portion of the bulb with your bare skin. Oils in human skin will damage the bulb. Handle it only by the metal base of the bulb insert.
     
   • 8Insert the new bulb in the appropriate headlight-assembly cavity and reattach the bulb retaining clip to lock the bulb in place.
     
   • 9Lift the assembly up and plug the wire harness plugs back into the respective blades of the bulbs.
     
   • 10Push the headlight assembly into its cavity, but before locking it in place, check the lights by turning on the headlight switch.
     

Common problem, there is a headlight relay in the power distribution center under the hood that needs changed out.

Things You'll Need:

• #H11-55W and #9005 replacement bulbs (low- and high-beam bulbs for 2007 and later models)
• #9006 and #9005 replacement bulbs (low- and high-beam bulbs for 2002-2006 models).

STEPS TO REPLACE HEAD LIGHT: Shut off the Camry's engine and set the parking brake for added safety. Also, make sure its headlights are switched off. Open the hood using the primary and auxiliary hood release latches. Locate the back of the headlight assembly and determine whether the low- or high-beam bulb needs to be replaced. Turn the headlight bulb socket counterclockwise to remove it from the assembly. Unplug the electrical connector from the back of the bulb. Carefully dispose of the bulb to prevent it from shattering or cracking.

1. 
   

look under the hood in the big fuse panel and look for High beam headlight relay the relay is more than likely bad

repalce the relay its going bad

Honda Headlight Bulb(H1) - This bulb is the high and low beam bulb for your vehicle. One bulb required for each headlight.

This headlight bulb fits:

• CR-V 2005-2006 (requires 2 bulbs per side)