You could have an internal short in your instrument cluster that is capable of backfeeding a ground through your headlight switch. If you want that diagram, let me know.
Check your cigarette lighter socket for pennies. The illumination circuit ties into the headlight switch as well, which a penny inside it will create a backfeeding ground.
If you have the Light-Out Warning indicator on your dash, check the Resistance Wire shown in the diagram.
I wouldn't do much driving until you get to the bottom of this. Shorted headlight circuits can cause fires in any number of places throughout the car. An 84 stang with 42k on the clock has almost surely been a victim of critter nests, and critter teeth.