Update...I replaced the resistor yesterday. Now, the fan works, and it works at low speed. Hooray. On the bad side, the fan works when it should not be working.
When I turn on the ignition, after the car has been sitting all night, before I even start the engine, the low speed fan starts running. When I start the engine, the fan continues running. It runs all the time. Now, if I turn on the a/c, the low speed fan stops running, and after a few seconds the high speed fan kicks in, which I think is as it should be. The low speed fan stays off while the a/c is on, even when the a/c high speed fan cycles on and off.
If I turn the a/c off, I have no fan at all for about 35 seconds, and then the low speed fan starts running again. This is when the engine appears to be cool enough that the fan should not be running.
It also seems that if the car does get warm enough that the low speed fan should be turning on, it instead turns off. It's almost like the fan is on when it should be off, and off when it should be on.
Help?