Cliff,
If you could share the specific model # we might be able to get more specific.
But generally the possibilities are bad heaters or bad thermostats. My self I
would not worry about 12V as that is the most inefficient and least effective way
to cool.
First, insure there is 110V at the back of the refer. On a simple system one wire
would go direct to the heater and the second would go to the thermostat and then
to the heater. By disconnecting either line to the heater and measuring between
the heater wires with an Ohm meter you should read about 50 ohms. If the heater
is correct the likely problem is the thermostat. Make sure it operates
mechanically. Then bypass it. If the refer then cools on 110V you have found the
problem.
Charlie