Thermocouple temperatures change drastically when my whole system is powered vs when just my arduino is powered
I don't see any earths at all in your pictorial sketch. If the output of the 24V supply does not have the output common with earth, the first thing to do is to ground the analog ground of your PCB. If that doesn't make sense to you, ground the (-) output of the 24V supply. 

If that does not improve things, and you cannot shield the thermocouple with a braid, try connecting each side of the thermocouple to the ground pin of the Maxim chip through a 100nF capacitor.