Need model # to help, seems like a bad cold control.

Thanks in advance, anybody that can help. I have a 10-year old Whirlpool side-by-side, that refuses stay cold. It'll run to get to the directed temperature, then shut off automatically then fail to come back on when it warms up. I have cleaned the coils underneath, made sure the fan underneath works, and replaced the defrost timer. Once the unit shuts down, I can cycle the Frige temp control to off then to the desired setting and the unit will come on and cool until the temp is reached then shut off again....repeat, until I forget or am gone long enough to spoil food. Anybody know what part is at fault?