2 Questions.
 

I bought a 90 GXL (so it says on the title) for $950.00, I was told it needed new U-joints, so I replaced the drive shaft. Now the car drives fine and everything but there is alot of heat coming from the shift knob boot. It even smells like something is burning every once in a while. I wanted to know what you all might think that is. My second question is, when I turn off the car, the headlights stay on, even after turning the switch to off. The gauge lights turn off just fine, but the headlights stay on, the only way i can get them off is to disconnect the plug right behind the dashboard. I wanted to know if any you may know what the source of that problem is as well.

The heat is normal. You need to take the boot of and replace the rubber booty. As old as are cars are they are sometime not even there anymore.

As for the smell, check for exhaust leaks, or coolant leaks With the rubber booty not there or leaking. You can smell allot thats going on in the engine bay.

Oh, and welcome to the forum!!

Sounds like the headlamps are gerry-rigged and you will need to trace out the wiring to see what is missing, bypassed, or faulty.