Update, well it seems like an issue with a relay. Any advice cleaning these types of relays? I have Deoxit D5, would this be advisable?
A chemical cleaner doesn't work so great on heavy oxide buildups.
I'd take the relay apart (cover off) and put a piece of paper in the contacts and squeeze the armature down to close the contacts. Then I drag the piece of paper through the contacts, like it's a sheet of soft sandpaper. You can see the black oxide come off, on the paper. Then, applying a dab of DeoxIT will add a film to prevent oxidization.
This is a trick an old telephone tech taught me, from the days of mechanical rotary relays in phone exchanges.
Careful you have the perfect tool to pry off the relay cover. The plastic gets hard with age and wants to crack. A screwdriver or blade with a thin edge works for me, and not over prying.
Make sure the pins (socket) are clean too, I use a chunk of pencil eraser or maybe a Q-tip with DeoxIT to rub off the oxide on the pins.
As far as the A/D (DAC counter) not stopping on mV, this is a difficult multimeter to understand.
At first I thought the Null Detector trimpot (R24 on 5524) might need to be adjusted. I think the Null Detector tells the counter to stop (latch/hold) the reading which is not happening. But this does not explain why other ranges are working.
Leads me to... I think the mV amplifier (Reference and Buffer Amp) has a problem or large offset on mV range, causing the Null Detector Output to malfunction or stay stuck. This whole multimeter is full of balance trimpots, so I would check it on Buffer amp with the ranges that are causing trouble.
This is quite the piece of art for 1967: 19" rack-mount sealed enclosure, 100,000 count with Nixies, fan cooling, runs to -55°C with the internal 400W heater, explosion-proof, 0.005% DCV accuracy... It must have cost a fortune in its day. It's serious MIL-spec.