Ground.
Winner
Ford
Weiner
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Ground.
Ford
I thinking that's a pretty good guess.
Since nobody has said it, I'm gonna' say.... rodents.
Doesn't sound like it's confined to the headlight(s).
Diagnosing electrical issues over the interwebs is just about impossible. I would start by working your way back to the switch.
Unplug the headlights, start the car. Turn on the headlights. Does the car die?
Remove the headlamp relay, start the car. Turn on the headlights. Does the car die?
Remove the headlamp fuse, start the car. Turn on the headlights. Does the car die?
That should narrow it down some, but without a wiring diagram, don't know how much.
Were you able to handle the repairs yourself?