My GE Profile side by side refrigerator is

Sounds like the damper or damper belt may be at fault. The cold air enters the refer compartment through the damper in the back. If that damper is not controled to close/open as needed it is likely stuck fully open all the time. That will let cold air in no matter what the t-stat says. See if you can view with a mirror and see if the damper "blinds" are fully open. If the t-stat is turned all the way down the dampers should close, as less cold air is needed. Then, as the t-stat is turned up, the dampers open to let in cold air from the freezer compartment. Let us know what you find and we can go from there.