All of the juice runs through the switch which apparently isn't designed to handle it very well. I installed relays on both the low and high beams and just use the headlight switch to turn them on. All high current now goes through the relays. I gained quite a bit of headlight output as well and my switch stays cool.
I used these sites for info and supplies and built my own harnesses.

suvlights.com Universal Heavy Duty Headlight Wiring Harness w/ relays for bulbs (9003/H4, 9004
Daniel Stern Lighting Consultancy and Supply
Innovative Wiring - Home
Installing Headlamp Relays - Dodge Diesel - Diesel Truck Resource Forums

 

I had same prob with my 94 and the dodge parts guy said the new switch was updated, but he just says what he is told. I changed switch and plug. no more prob yet! but i think but i may be wrong that there is already a relay on each light at fender. corect me if im wrong. if not ading a relay would be a huge upgrade. and very cheap and easy. and if your switch goes out you can manualy turn on if need be.

 

THere are alot of good suggestions on here. You could always just clip the clearance light wire from the back of the headlight switch and run it through a seperate switch. no more problems.

 

My wires burned up.... not the switch itself...

 

A friend of mine had the same problem. His was an 01 though.

 

Wow guys you are very helpful! Going to try a new switch first but if I have the issue again I will just instal a relay.

 

when you do make shure you get the updated switch and change any bad conections on plug. If you did not know you can remove the wires in plug and only change the one that got to hot. if you dont fix plug it will happen soon again.

 

hi
do yourself a double solid and get one of the wire harness upgrade kits with relays.

you will lower current draw on headlight switch and gain brighter headlights .

well worth the money !