I have a 66 mustang with a v-8 and the problem is that when I went to turn off the engine it continues to run like it was never turned off. I’m not talking about dieseling,etc. I replaced the ignition switch, starter relay and starter. This engine has about 2k on it. It has just done this again but doesn’t all the time. Once you get the engine to stop it starts and turns off normally.
You have a bad ignition switch in all likelyhood. It’s the only link to the ignition system on a 66’ unless you have modified it in some way.
There are two hot wires to the coil. One, a direct 12V lead, comes from the “start” terminal on the switch. This wire also engages the starter solenoid, and could be back-fed from it.
The other is a resistance wire coming from the “on” or “IGN” terminal of the switch. It is what supplies 6 volts to the coil after the engine starts.
The next time this happens, remove these wires from the coil (engine will stop) and see which one is “hot”…Find the short and fix it…
Sometimes the short is easy to find if two wires are fused together and discolored. I pried two apart under the dash once and fixed a turn signal problem. I had an NYC car and you couldn’t see anything under the dash until you removed leftover stereo wires from the last three stereos that were installed. Then there was the excess Bondo problem which I never fixed.