1999 jeep cherokee-hooked up to a trailer with electrical problem-I instantly lost dim headlight and both driver's side running lights. I suspect the running lights module. I found the replacement part replacement part but cannot find it's location nor the location of running lights relay. I have checked bulbs and sockets for continuity and that is good-but no voltage...

If you have modified the wiring then I would put wiring back to how it should be

Check ground wires. One for each head light. Mounted to the body, close to the light. Drivers side has lost ground and is try to get it threw the passenger side. This is why it is dim.

Headlight switch is what it sounds like.

My first guess would be a ground problem.
The headlights and tail lights are controlled by the same headlight switch, but the brake lights are on a totally different circuit.
You may have to get a shop manual to get the wiring diagrams needed to troubleshoot the circuits if the ground connections are ok.
The lights involve the body computer and other components.

The circuit load is much higher with the HID lights, you need to rewire the headlights to a large gauge wire.

Are you using the special heavy duty bulbs required for this model.

SOUNDS LIKE THE PROBLEM IS A CORRODED SOCKET.REMOVE THE LENS AND BULBS.SEE IF THERE IS EVIDENCE OF CORROSION AT THE CONNECTOR AS WELL

TEST
remove both bulbs from the lamps but leave them on their wires switch on and see if the bulbs are still different brightness if not they the passenger side lamp housing has damaged mirror is dirty or missalined and needs adjustment. if they are different brightness then this may be a earthing problem.

Probably a stupid question, but does the car start? Have you checked the battery for proper voltage? (and charge rate). Other than that, I'd suspect a poor ground somewhere causing excessive resistance & draw on that system.(headlamps draw far more than dash lights so power is used up there first)

Check for a broken 12V+ or broken ground wire in the rubber boot that goes from the drivers door to the hinge pillar. Had this problem on mine and found both wires broken from years of flexing when door opened and closed. Don't try to splice the break, as the wire is too short and is also brittle. Run new (same guage) wire through the boot and splice inside the door and under the left side kick panel.