I finished soldering together the other two channels for temperature control. This is what the board looks like now:
I began troubleshooting the new channels. The second channel (top right) is tentatively working up until the buffer stage, but showed some strange behaviors. When it is working up to the buffer stage, adding the buffer stage drive the output of each stage (and all of the corresponding inputs) to about +12V. When the buffer stage is removed, everything remains at +12V, but this is sometimes fixable by toggling the power, and consistently fixable by just pulling out the OP27 chips and putting them back in. As far as I can tell, everything is powered correctly, and nothing is shorting or disconnected where it shouldn't be.
For the third channel (bottom right), only the AD620 stage is working. Adding the next stage causes both the outputs and the inputs to go to +12V as well. The connections here also seem correct as far as I can tell. I'm thinking it's possibly related to the problems in the 2nd channel, maybe there's some buildup of charge, or other hysteresis, somewhere, that is messing things up. The observed behavior of this third channel is consistent enough that it may just be a connection error somewhere. The inconsistent behavior of the second channel, though, suggests that it probably has to do with something transient.
Will continue troubleshooting when I get back.