New Board. New Thermister. Still Freezing

When you turn fridge OFF ALL DC power to control system is shut off so the only way the main flame (no pilot flame) is staying lit is because gas valve is not closing.

Your thermistor and lower board where not the problem......gas valve is.
The 'auto thermostat' (non adjustable pre-set at factory) is what controls fridge cycle via temp sensed by thermistor. The thermostat turns heating source (electric or gas) on/off to control fridge temp.

You can make minor temp adjustments (6*-8*F) by changing position of thermistor on fin.
Move thermistor clip up for cooler
Move thermistor down for warmer

BUT with gas valve not closing.....main flame continues to burn causing excessive cooling

Time for a new gas valve....part #2932615012 About $125

It sounds like the gas solenoid/valve is stuck. First see if there is 12 volts trying to open the valve when the box is cold. If no 12 volts the flame should go out until the box needs to cool again. If the flame stays on then you can give the valve a whack but it probably needs replacing.

Tha fact that it works ok on AC would indicate the board and thermister are good.