why would the headlights work only with the car running, all the other lights work without car running, 1977 firebird

My guess is that someone in the past has rewired the original circuits. Get a wiring diagram and you can change them back if you wish.

I seem to remember that at least one of my Dad’s old 1970’s Chevys had something like this. The lights wouldn’t work when the car was off, or the radio would only work if the lights were on, or the lights would only work if you put on the hazard lights and stepped on the brake (which caused the hazards to stop flashing and stay on steady.)

Something weird like that. (I was 9.)

Anyway, if I were you I would get in contact with a Firebird owner’s club, I’ll bet there are a lot of fans of the old birds. Ask them how it is supposed to work. They might even share the manuals.