You mentioned your headlights were going off/on as you drove. The supply wire could be grounding itself out (which definitely would cause the wire/connector to get hot until the thermal protection cut the headlights off). Then the thermal protection would cycle the lights on for a bit/off for a bit as long as you drove. To check for ground, get a handheld electric tester. Set it for the diode checking function. In the first image below, I've done that. Remove the headlights and see what the resistance in the wire leading from the switch to where the headlights should be is. How? Connect one of the tester leads to ground and the other to that supply wire. If everything is ok, the reading should show as a whole number...maybe something like 3.3 or some other whole number (as you see below in the second image). If that wire is connecting ground, you may hear a buzz (if your tester has that function) and even if you hear nothing, you'll see the reading go to all zeros (as in the third image) and some meters may show you some very small number. If you see all zeros or a very small number (like 001 to 007) and you have the tester set for the diode function, your wire is grounding itself. I've added 2 images below showing a really cheap tester I got free at Harbor Fright years ago as image 4 and 5 (4 shows wire not grounded and 5 shows wire grounded). Good luck.
~ Jon 1952 1/2 ton with 1959 235 | T5 with 3.07 rear end
