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They are replacing both parts just to make sure that the problem is fixed since these are the only 2 parts that can cause your problem. As for how to diagnose between the two, you'll need to remove the thermostat and test continuity between the 2 terminals on the thermostat for the refrigerator. If you have continuity through the thermostat while the unit is too cold, then the thermostat is bad. If you don't have continuity, then the damper isn't closing and you need to replace the damper.
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If you had a defrost issue, the evaporator behind the back panel inside the freezer would be completely packed with frost and the refrigerator would be too warm, not too cold. Your issue would be the operating thermostat the $200 part apparently.