Well... slightly premature in my previous post. I checked the unit's AC, DC, Ω and A the first evening and all was well. The next day, I rearranged my bench to put all the DMMs in one place and let everything warm up so I could check variance. The Keithley started giving weird readings - the dB annunciator lit up sometimes, it showed V when reading Ω, gave a steadily increasing reading on ACV, etc. Then I noticed the mode selector switches were sticking - this correlated with the weird readings most of the time. My guess is that bringing the meter from its warm environment into my cool basement caused some of the lube on the switches to become more viscous.
I took the board out, tilted it so that any gunk would run out the front onto a paper towel, and hosed the switches down with 91% IPA. Now it worked. For a while... and then started sticking again. Repeated the procedure... exact same thing happened. So I had to run out today and buy some high quality contact cleaner/lube. This time I left nothing to chance; hosed the switches down, made sure nothing ran up the board into the components, let it dry, hosed it down again, repeat. Exercised the switches in between applications.
So far (it's been on for a couple of hours) it's working at what looks to be appropriate accuracy - tracking the Fluke meters all the way up the scale on AC and DC. I did a spot Ω check and it looked good (in fact much better than Saturday), but am going to repeat that later after I've thoroughly examined the AC/DCV ranges.
So, a question if anyone feels they can contribute: the stuff I used is a commercial product which specifically states it's a cleaner as well as a lubricant - I think it's Puretronics, which was about all they had. Is there a need (or value) to hit the switch contacts with Deoxit after it cooks for a while? Or should I quit now while it's working?
Oh, in fact one more question: I took the IEEE-488 board out and left it disconnected while testing the unit. I really don't need the capability; is there any downside to just leaving the board out and plugging the holes in the case? I can't think of any.