You also might want to hook up a relay between your switch and headlights using your battery as the power wire to the relay. The reason the switch overheats is because that switch was poorly designed and the load is too great for the wiring. By using the relay, it takes the load off of the switch. It is a very common problem with '87-'93 Stangs. I had the same problem and after installing the relay, I never had the problem again after 7 years with it installed.
Just a thought for ya.
I'll look around and try to find the wiring diagram if you're interested!
Glad you got it working, though!