Alot of good points here by many members. Here's my thoughts:

Disconnect your battery. Then take the time fully inspect your wiring. Anything look like it was altered, "repaired"? Feel the wires, do they bend easily? My guess is the original wiring is in place with some issues. 50+ year old harness is a fire waiting to happen! All the harness's are available from all the parts companies a out there. They are plug & play, built to original specs, correct connectors, etc. I am replacing all of mine on my 1968 Firebird. And of course, check all ground, including instrument cluster/dash. It is a common problem.