I got it fixed! Now the car has two working headlights instead of just one! The stupid connector is much bigger than the old one though, so you have to install/remove the connector from the headlight while its still in the housing. Otherwise it won't clear the lip of the hole...
I didn't see part of this answered, so here ya go...
Bad/loose/dirty connections can add substantial resistance. Resistance causes heat, and so enough added resistance could melt the connector possibly.
Did it lose connection because of the burnt connector?
I've seen this before on GM's. Get a new pigtail/insulator for the headlight and wire it in.
Of course, pay attention to which wire goes where. I've repaired several of these and
installing a new connector takes care of it.
I fixed the problem for free. Found a foot switch up by the firewall, that controls the high beams. It was turned on. Funny thing is, once i turned the foot switch off, the turn signal stalk works correctly and I can activate/deactivate the high beams