I cleaned the grounds near the headlights, took out the turn signal and side marker light bulbs and sandpapered them to clean contacts.
I still have the same problem. The front and rear turn signals work when the headlights are turned off, the rears work with the headlights on, but the fronts do not flash when the headlights are on.

Both signal indicators in the dash blink when the signals are turned left or right and regardless if the headlights are on or not. The hazard flashers work front and rear with the headlights off or on.

Its certainly possible that wires are hooked up wrong, but if so which ones should I be looking for?

This is a 73 with a six conversion in it. The car was dormant for over 10 years and I just recently brought back it to life.

You have either the turn signal buckets or the side marker lights hooked up wrong. One combination of wires-in-the-wrong-place will do this; I forget which one, though. (Ground swapped for parking-light wire?)

I suggest pulling the bulbs and using a continuity tester to check to see which lug on the back of the bucket goes to which contact inside the socket. Match that up with the wire colors on the wiring diagram.

--DD