I looked at the temperature sensing board that Andrew and Kira were working on (PSL:1889) that had ~12 kHz ringing when the op-amp was driving a 100 Ω load or smaller. I saw the same ringing except that it was ~830 kHz.

Since this ringing is likely caused by the load drawing too much current from the op-amp, Andrew suggested removing the second op-amp stage and adding a resistor in series with the output of the first op-amp; schematic for a single channel attached. After adding a 1 kΩ resistor, the ringing stopped.

The board Andrew and Kira were working on was having intermittent issues with the +12 V regulator which they had replaced; it looks like one of the pads could have been damaged. I modified one of the other temperature sensing boards to use this new circuit for all four sensing channels.