I think this is a case of "when you hear hoof beats think horses, not zebras." I think the motor was extremely hot because it was working too hard raising the lights. An hour later, with the battery disconnected, the motor was still quite hot .. all that copper. Maybe the bearings for the headlight "rod" are sticking or maybe the small gear box is jammed in some way.
The circuitry appears OK .. lights go up and down with the switch and stop when they should. The lights themselves go on and off. The lights go up slower than normal and the gears sound like they are working hard suggesting a lot of friction somewhere. I'm going to pull the motor and gear box and see if anything is stuck or jammed. Hopefully the motor hasn't burned out. But first I will reconnect the battery and see if the motor heats up without me trying to operate the headlights (no touching the headlight switch). If it does then there is a circuitry issue.