The refrigerator thermostat (also called a cold control) is what controls the temperature and it tells it when to run and when to turn off. It sounds like that part is most likely bad. You can unplug the refrigerator and remove the thermostat. With the thermostat turned to the off position, it should not have any continuity, if it does than the thermostat is bad and should be replaced. Also, an easier test that sometimes will confirm the thermostat is bad is to turn the thermostat to the "off" position, and then tap on the knob with your finger. If this causes it to shut off, then replacing the thermostat should fix your problem. Let me know what you find.
This is the thermostat for your refrigerator:
Cold Control - 2198202