Evaporator is typically inside the freezer, in the back. Not the center partition. Look in with a flashlight, and see if the back inside of the freezer has "snow" accumulating. Please let us know what you find.
Check to make sure you didn't change the thermostat setting inadvertantly.
I agree that the thermostat is the most likely culprit, but I would check to make sure the fridge defrost timer isn't stuck in operating mode before you replace the thermostat.
Basically, the brains of a frost free fridge is the defrost timer. It's job is to divert electrical power away from the thermostat for about 20 minutes or so once every 20 hours of fridge operating time. The power is diverted to the defrost heater to melt all the ice that's accumulated on the evaporator coils.
But, it's only a tiny little motor in the defrost timer, and a bit of friction can cause it to get stuck; either in defrost mode, or in operate mode. If it gets stuck in defrost mode, then your fridge essentially goes into permanent defrost mode until the defrost timer unsticks. If it gets stuck in operate mode, the fridge will operate continuously without defrosting itself, and eventually frost will start forming inside the freezer compartment.
Look on the partition wall between the fridge and freezer sides of your side-by-side for a removable panel. Behind that panel will be the evaporator of your fridge, which is the source of all the cold the fridge makes. It should not be clogged up with frost. You should have anywhere from no frost to a uniform coating of frost over it's entire surface. If it is all clogged up with frost, then the fridge is not defrosting properly, and that could be due to a stuck defrost timer.
If you can see that the evaporator coil isn't clogged up with frost, than I'd go ahead and replace the fridge thermostat.
PS: Nowadays, the correct term for a fridge "thermostat" is a "cold control" because it serves a different purpose than a true thermostat.