The temperature sensor reading is being railed to the negative power voltage whenever a new input command is entered:
The first ramp is at the output of the instrumentation amplifier AD620, the second ramp is at the output of the low-pass filter Andrew put in recently. Also, I looked at the signal going into the AD620 and it did not exhibit this drop, so the problem comes from the AD620 circuit. I tracked the current being drawn on the display of the power supply, and never saw it go above 250mA, but I'm thinking if it's a current problem (overdrawing), it could be happening fast enough to not register on the power supply.
I plan on using a resistor and the oscilloscope to track the power current more closely while I change the input command to see if it is the problem.
Note: if the leads on the temperature sensing circuit are flipped such that the signal is negative, the signal rails to the positive supply voltage whenever an input command is entered.