High beam indicator does not work when the engine is running

It sounds like the alternator is throwing AC voltage and you remove the belt or the wires and recheck? If you do the wires they are hot so be careful with them to make sure they do not touch ground.

Please report back.

Cheers, Ken

Thanks for replying Ken! I removed the alternator belt and for a few seconds after starting engine, the high beam indicator and odometer stayed on, but then went out and will not come on while engine is running, unless I mistakenly thought they were on for the few seconds. I put belt back on and am studying the wiring diagram. I misdiagnosed a bit, the high beam ind and odometer stays on if key is in first spot, but they go out in "run" position, engine running is irrelevant. I am guessing something with DRL Module or Gauge Cluster Ground in Run position?

This sounds like a BCM issue. I would remove the electrical connector from the BCM and check it for corrosion. Here is a wiring diagrams for the headlights so we can see how the system works and the BCM location. Check out the diagrams (below). Please let us know what you find.

Cheers, Ken