Check your current draw across your various fuses. Feel for heat on any of them. Especially the one in the headlights circuit.

Then check your battery/alternator circuit.
Should be over 12V with engine off. And 14.4 (STEADY) when the engine is running. Test this when the headlights are on and when they are off,it should still be 14.4V (This tests the regulator).

Sounds like an excessive current draw somewhere in the system. Does your coil get REALLY hot? So much that you can't touch it?
It could be the coil is dying.

I agree that the symptoms are VERY strange. Maybe the headlight relay is shorting out when the lights are turned off? Hard to tell.

I hope this gives you something to think about.