Does the fridge shut off in the AC mode, as well?
Before replacing the control board, replace the temperature sensor that's located in the interior lamp fixture. The part number is XXXXX and you can buy it at any Norcold authorized servicer(most RV dealerships).
It's much less expensive, and it's easier to replace if you cut the wires, leaving enough to connect to the replacement part with small wire nuts.
If that does'nt solve the problem, you'll need to check the connections on the control board. If they're clean and tight, you'll then need to replace the control board.