So I called Casita
and finally got the problem diagnosed figured out: the little plastic tab on the switch that breaks the connection when the thermostat is turned off had snapped off. Not obvious!
Of course, I can't just replace the plastic switch top, but need to replace the entire thermostat (or just keep sticking a piece of rubber or something in there to break the connection. Kind of a pain -- but cheap!).
As suggested, replacing it with a digital thermostat sounds good. But first off, to remove the old one, I will need to drill out the rivets that attach it to the wall. Is that no biggie? And then the new one, I guess, would likewise be riveted to the wall, yes? I do have a rivet gun -- which I have yet to use.