Looks like this is a non-issue, as my water heater is now being fixed.
I'm having some tile work done in one of the bathrooms, and the tile guys managed to run a screw into one of the pipes in the master bathroom. They ran downstairs to shut off the water, and then called in a plumber whom they usually use.
The plumber fixed the damaged pipe, then went downstairs to turn the water back on and saw my water heater half taken apart. They called me to ask what was up with it, and I explained the problem. The plumber said, "Heck, I've got a thermostat that'll fit that on my truck. Want me to go ahead and fix it?"
I said yes, so now for $100 it's being fixed (and tested) as we speak.
Thanks for the advice everyone. Hopefully I won't go home to any water damage tonight!