I verified that the problem was the dimmer module. I pushed it out from behind the dash, and found that the previous owner had 'fixed' the broken dimmer module by bridging the contacts at the back with a paper clip. This jiggled around and sometimes didn't make contact. It also generated a lot of heat, probably was a fire hazard.
I was able to disassemble the unit by removing the black plastic ring and cover around the outside end, then pull the innards out of the module by the square metal tab. I found that the power transistor (attached to the metal backing) was obviously blown - it was split and melted.
My simple idea was to repair / replace this. To my surprise it worked ! Here is the basic procedure for you people in similar situations.
To get to the transistor, I released the black plastic clip in the center of the circuit board to separate it from the metal backing, then de-soldered the power transistor from the board. This allowed the two big pieces to separate.
After a lot of research, I decided I needed a NPN transistor. I went down to Radio Shack and found one for $1.69, soldered it in, and to my surprise it worked!
For you folks wanting to do this yourself, the transistor was a TIP31. It has a TO-220 case, NPN silicon, Radio Shack P/N 276-2017.