The repair.
Since the meter acted funny on the range switch, that's where I started looking. It turned out that the little pads on the pcb are spaced very closely together. Normally that's not a problem, but now the range switch had scored and pushed the pads out a little, and they could intermittently short circuit creating the strange reactions to the switch.
The problem was solved by making some cuts between the pads, you can see them in the photo.
Why the pads had been squashed out I'm not sure, but I did find that the range switch should clamp in the hole in the pcb, but one of the clamping tabs had broken off (also visible in the photos). I don't know if the person who was in there before me had caused that or it had left the factory that way. With that broken off clamp, maybe the pressure on the pads isn't even and some get scored and squashed. They weren't all as bad.
A cheap fix, hopefully a long lasting one.