Eventually, I decided that I'd had enough, and decided to buy a new programmable thermostat. It should pay for itself fairly quickly in energy savings, by virtue of being able to lower the temperature a few degrees while I'm asleep or at work, automatically, without my having to remember to lower it. I called Bishop's to ask if they had any in stock, and they did. I ended up with the same model seity1 got last month, a Hunter model 44110 "Set & Save" that has 5+2 programmability... the set of five weekdays and set of two weekend days are programmed separately. A more expensive model has 5+1+1 so you can program Saturday and Sunday separately, but neither that ability nor the Indiglo nightlight seemed worth the extra $14.
So, my new thermostat is now installed. It was quite easy to do, and not surprisingly, I discovered that whoever put it in hadn't bothered with drilling or putting in a wall anchor; the old thermostat was just screwed straight into the wallboard. There's a small area with slightly darker off-white paint peeking out behind the new, narrower unit, but it shouldn't be too noticeable.