What could cause the cooler to fall below freezing temp other than a faulty thermostat?

probably a faulty thermostat as changing settings should make a difference
with fridges ( beverage coolers) the lower the number the colder the setting
have it checked by a technician before making the replace decision

Improper refrigerant charge. Its critical and measures coil temp and not cabinet temp. If you did add gas that is the answer. Take out all the gas, put vacuum pump on it, then weigh in exactly in ounces the correct charge, its listed on the spec. tag. Hope this helps and good luck.

You killed the messenger! The refrigerant charge is probably the problem. The thermostat senses the coil temperature and not the cabinet temperature. If somebody put gas in it, the gas has to be removed, a vacuum drawn on the system and then the refrigerant charge weighed in by the ounce. It takes a critical charge and needs to be weighed in by an accurate scale that weighs in ounces. Hope this helps and good luck.

Is it a digital temp. display? Does this show a low temp? It should show +/-34 degrees. Listen if the compressor runs continuously. It should run for a while and then switch off. When you change the temp setting, as you turn it down, the compressor should stop. If it runs continuously then the likely cause is a stuck/faulty thermostat.Before replacing this, switch off and leave open until unit is at room temp. Then switch on again and monitor. If the thermostat if faulty and the compressor runs continuously you can damage it by overheating. The compressor will run for a good few hours after restart, until set temp. is reached and then switch off automatically. Good luck!

Try turning thermostat knob all the way to left or counter clockwise. This should shut the compressor off. If the compressor continues to run the thermostat is stuck closed & bad. If the compressor remains off with thermostat turned all the way to the left try turning it slightly to the right (warmest setting). See if it stops freezing. If it continues to freeze set at warmest setting replace the thermostat. Try pricing a thermostat out at Refrigeration Hardware. Link attached. Refrigeration Hardware Supply

If you manually reset by pressing overtemp thermostat, then check your operating thermostat.. maybe over heating.. if so replace thermostat.. if pilot cuts out completely prematurely without reaching set temp then replace overtemp cut out thermostat as it is faulty.
Cheers

This uses a temperature sensor and control board.

Vino temp @ 1-800-777-8466

The "clunk" noise sounds like the internal compressor motor mounts have broken.

is the cooling system full,if so you may hve to get the heater core cleaned as they can plug up if the dexcool anti freeze has not been changed,could also have defective thermostat(stuck open)if you have gauge does it go half way or stay low temp,other causes could be stuck air blend door in heater system

You have an automatic defrost problem. Here are the three components that can cause this defect in order of most common to least. #1 defrost timer or equivalent adaptive defrost control board, #2 defrost termination thermostat (should be replaced regardless since it has been frozen in ice several times) #3 defrost heating element(rarely fails but is possible). If you have a meter you can check the thermostat and element for continuity they should show an infinite reading if they are operational. the termination thermostat must be below 35 deg f to show a closed circuit.

Could be the thermostat isn't working and the unit is just running and running all the time. To me that seems the most simple explanation right now. Can you check the temp inside when on the lowest setting for a day, then turn it up to the highest for a day and see if there is a huge difference in temps? Does the compressor motor ever turn off or does it run continuously? Suspect that t-stat...