It sounds like one or both of the headlight connectors is mis-wired, using the ground pin of the 3-prong connectors as either high or low beam. That would allow voltage to back up through the indicator light circuit and keep the light on all the time, while still allowing the headlights to function almost correctly. I've seen this happen occasionally when a new wiring harness was installed by someone without a lot of experience at rewiring a vehicle.
Jerry

"It is better to be silent and be thought a fool than to speak and eliminate all doubt!" - Abraham Lincoln
Cringe and wail in fear, Eloi- - - - -we Morlocks are on the hunt!
There is nothing noble in being superior to your fellow man; true nobility is being superior to your former self. - Ernest Hemingway
Love your enemies and drive 'em nuts!