Headlight assembly switch does not work properly

There's not likely a relay involved in a dome light circuit. Did this problem start after you changed the headlight switch? I would be looking for a wrong, shorted or swapped connection on the headlight switch (assuming that's where the dome light is turned on from).

Might be a short in the multifunction switch.

Have you checked the switch?

This means that the dome light is getting power and grounded by the jam switch. by turning the headlamp switch it has a detent position once there this should ground the dome light if it doesnot work the grounding wire from switch to body is loose or broken.

the switch on your dome light is more common problem.
remove the switch and inspect for burn marks and corrosion
clean and install
the switch will have 12V Battery power at all time.and ground from the door switch.
second ground will be on the switch and clowse the circuit only with switch to ON position

• Vehicle Repair Guides I would check the switch.
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Headlight Switch

ABCDEFGHIJKLMNOPQRSTUVWXYZ The headlight switch controls most of the vehicle's lighting systems. Many complaints concerning the lighting systems can be the result of a faulty headlight switch. For example, dim or no instrument panel lights, dim or no headlights, dim or no parking lights, and improperly operating dome lights can all be caused by the headlight switch.
Dash-Mounted Switches
A headlight switch. You'll need to remove the switch to test it. With the switch removed it can be tested for continuity and the connector plug will serve as a test point for the lighting circuits. First test at the connector.
In this procedure the terminals are identified as follows:

• Terminal B = battery
• Terminal A = fuse
• Terminal H = headlights
• Terminal R = rear park and side marker lights
• Terminal I = instrument panel lights
• Terminal D1 = dome feed light
• Terminal D2 = dome light

Consult the service manual for terminal identification for the vehicle you are working on. If this is not listed in a separate chart, you should be able to identify the circuits from the wiring diagram.

• Connect the 12-volt test light across terminal B and ground. The test light should light. If not, there is an open in the circuit back to the battery.

Testing for battery voltage to the connector. Reprinted with permission of Ford Motor Company.

• Connect the test light across terminal A and ground. The test light should come on. If not, repair the circuit back to the fuse panel.
• Connect a jumper wire between terminals B and H. The headlights should come on. If the headlights fail to turn on, trace the H circuit to the headlights. Also check the ground circuit side from the headlights.
• Connect a jumper wire between terminals A and R. The rear lamps should illuminate. If not, trace the circuit to the rear lights. Also check the ground return path.
• Connect a jumper wire between terminals A and I. The instrument panel lights should come on. If not, trace the circuit to the panel lights.

The chart below indicates the test results that should be obtained when testing the headlight switch for continuity. use an ohmmeter or a self-powered test light to test the switch.
Steering Column-Mounted Switches
On some vehicles it is possible to test the steering column-mounted switch without removing it. The test is conducted at the connector at the base of the column. However, on some vehicles it is necessary to remove the column cover and/or steering wheel to gain access.

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Just talking about tail lamps, did you check the fuse? Not only check the fuse, use a test light and check voltage on fuse circuit. At the tail lamps, the brown wire is voltage, headlamp switch on. The black wire is ground, check them both. You can check ground with test light, one end of test light touch battery voltage, the other end touches the ground wire terminal, if the light comes on, the ground is good enough to turn on the test light.
Usually the first thing you do is check voltage and ground on this circuit before changing parts. Just my opinion.

check both, and the wire had to be a hot wire not a ground,a ground wire cant do anything unless you touch it to apower source, anyway check both boxes notelling what other fuses blew, better save than soory. good luck.

Check all the bulbs at tail end for parking and brakes, test grounds and harness for trailer. Check fuses. clean batery terminals, Re instal head light swich properly and make sure all wires conect.
Vitally

I would remove the dome light, and disconnect the wiring connector, and then put in another fuse. If it does not blow, I would think that the dome light switch is shorted to ground.