This is based on the wiring diagram for my 68 Charger, but should be accurate for these issues.
The dash lights pick up power separately, it's kind of goofy. The headlight switch has a black wire the goes to the dimmer. That wire feeds the rheostat inside the dimmer that dims the instrument lights. From there, a tan wire runs over to the fuse box. The fuse box has a separate fuse just for the dash lights. The output of the fuse is an orange wire that routes around the car feeding not only the instrument lights, but the cigarette lighter lamp, the back of the instrument panel plug, individual dash lamps etc. I would think it's actually the headlight switch or wiring at fault.
Did the turn signals every work? Power to the switch goes through the flasher first. A black wire from the ignition switch is key hot. The red wire from the flasher feeds power to the turn signal switch. If you jump the flasher unit plug, the turn signals should light up when you move the switch, but they won't flash. This trick might overcurrent the TS switch so if you try this, don't leave it on for more than a short time to diagnose power issues.
Tim