Burnt turn signal switch 68 Valiant

What do you have for a fuse in the circuit?

I would examine the sockets very carefully for damage in the contact mechanism. If for example the springs in the sockets somehow contacted one of the bulb contacts, etc.

Next check the harness from the lights all the way to the kick panel. You are going to have to remove the rear seat, and pull up the door sill, and remove the kick panel. You of course are looking for pinched / damaged / bare spots in the harness.

From the kick panel to the switch is "normal" under dash wiring. Unless you want to untape it, it is actually unlikely that something went wrong there.

It IS possible that it's something in the switch itself. There are two things that cause "meltage" and sometimes, they are related

One of course a short causing very high current, and the second is a bad connection either at a terminal or crimp or rivet, or bad switch contact.