Go through your dash wiring harness and clean your grounds. There are multiple throughout the 14401 harness attaching at various points on the dash steel frame, IIRC is one near the light switch attached to the metal frame of the dash, there is also the steel support rod at the center of the dash to the firewall and then there is one on the right side. Those and the radio retainer clip.

But...

I installed a new 12 gauge wire that I soldered onto the ground wire at the point nearest the light switch, connected that lead to several of the other grounds, ran another lead to the tail lights and tied in on those grounds at the left rear, and combined all those leads at the dash and ran a single 12 gauge black wire out to the negative battery terminal. For that I replaced the negative battery cable with one from a '72 Mustang that had the additional 12 gauge wire integrated in the battery clamp and crimped that wire using the supplied crimp type connector. That solved a multitude of intermittent electrical issues in my '72.

Your light switch getting hot is a sign that you are having a ground issue. I ran my '72 GT 4 Speed with that issue quite a while and eventually cooked the light switch. Since I added the direct ground lead out to the battery (this was about 25 years ago) I haven't had any issues since then. OBTW, during the current restoration I have actually added 7 steel blocks welded to the chassis in various places and have strung a ground lead that interconnects the doors, back body at the tail lights, at the cowl for interior and engine side ground straps along with the fenders and headlight / radiator support. Every electrical issue I have had with that specific car has always resolved to a bad or disrupted ground. JMHO