Got it to work. One of the connections was faulty. I decided to check the temperature measured against a thermometer. The sensor showed 26.1 C, but the thermometer showed 25.8 C after I let them both cool down after heating them up. The temperature of the thermometer was dropping at the time of measurement, but the temperature of the sensor was not. This is still a rough version of the final sensor, so I'm not sure what exactly causes this discrepancy.
Tried taking the circuit from the breadboard to the PCB. I attached all the components to adapters that would allow them to be connected to the PCB. From the first picture, the first component is AD586, the second is AD590, and the third is LT1012, along with a resistor across it. I then soldered the connections between the components, as can be seen in the second picture. When I tested out this version of the circuit by hooking it up to the DC source, I got a reading of ~-15V. I will have to check all the connections to make sure there is contact where there should be one, and no contact where there shouldn't be. I had issues attaching the tiny AD590 and LT1012 to its adaptor, so the issue may lie there as well. I'll also check that each component is in working order as well.
Once I figure out where my error is, my plan is to build two more of these and place a metal object such that it contacts only the surface of the AD590s. This would allow me to compare the three values to the actual temperature of the metal, which would then tell me how accurate this setup is.
Note on the resistor: I measured all the resistors and chose three that had exactly 10.00k Ohm. The voltage detected is dependent on the resistor, so if we are to take three identical copies, I ensured that there would be no error due to the resistors being a little different.