I was able to check the voltage of the 24V leads to the solenoid and it fluctuated when it clicked, so the solenoid appears to be fine. I tried to find any kind of other sensors or switches and didn't have much luck. I traced the 24V leads (as best as I could) back and they go into a box behind where the thermostat lines connect. I opened that up and noticed that they connect to a smaller solenoid (about half the size of the other one). There's also a transformer on top of that box that wires connect to.
After poking around, I turned the breaker back on and a few minutes later, the heat came on and it started clicking again. Rather than turn the heat down on the thermostat to shut it off and stop the noise, I tried turning the heat up hotter. When I did, the clicking indoors stopped. I came downstairs and heard the fan of the outdoor unit and it sounded like it was turning on and off slightly (almost like it was having a power dip). I went outside and heard a similar clicking from within that unit. I had opened that up earlier just to see what was in it and saw a solenoid, but nothing that I recognized as a temperature sensor. It did have a circuit board that everything connected to. I think one of the leads from the solenoid connected to the board and the other went to one of two large cylindrical capacitors. Another time, when it was clicking indoors, I listened to the outdoor unit and it was going on and off (and probably clicking as well) at the same time. It would seem like something is making it happen in both places.
I also noticed that the front of the box that the thermostat lines screw into has two lines connected to each of the color coded terminals. One line to our main thermostat, and another that runs to our outside unit. I'm not sure if that makes a difference, but thought I'd mention it. That box seems to be the common junction point where the indoor and outdoor solenoids connect, so maybe the small solenoid in there is at fault.
Is it possible that the new temperature sensor could also be bad * I don't know what the resistance should be, so I can't test it to be sure. I tried looking online, but no luck. Thanks again!
I was able to scan in a the air handler and heating unit schematics, which you can see -
The picture is - w w w .fataldelay.c o m SLASH Trane_Schematics.pdf