Just a quick note to anyone else ever having issues with Thermocouple module reading inaccuracy and temperature fluctuations. For us the issue turned out to be that someone had substituted regular copper wire for real thermocouple extension wire on our calibration set up. They even used the correct color code for the type of TC wire we were using.
Double check your connection wiring and make sure it is the correct type of thermocouple extension wiring to match the thermocouple type or calibrator output you are using. This wire is typically stiffer and the metal in the two leads often looks different than the other and than copper wire due to its different combination of metals.
Each module has its own cold junction compensation temperature sensor inside the module right next to each plug input. Since the modules do tend to heat up 5-15 deg F when sitting at room temperature, that can warm this cold junction up, even compared to a calibration source right next to the module. Without the correct connection wire, this creates two more junctions at different temperatures which create a temperature offset reading equal to the temperature difference between these two junctions.