I got a used one for free that had the same problem. Took it apart and troubleshooting the power supply board (the big yellow one) and it wasn't converting the AC power to DC. Original owner also said it would run with the DC cord but then that went missing.

Got a cheap replacement board ($9) that's similar enough from AliExpress and replaced it. Powers up and verified it's now outputting correct DC voltage and the cooling element gets cold. The big heatsink would get pretty hot without having the fan blowing on it, and the cooling module stops getting cold.

But after putting it back together the inside never actually gets cold. Seems like it's either not pulling heat out of the inside of the fridge, or it's still overheating even put back together with the fan blowing on it, and not able to cool down.

These little units use a 12volt DC peltier module that transfers heat from one side of the plate to the other, so the cold side gets really cold and the hot side gets hot, and then needs to reject that heat into the heat sink.

I might see if reapplying new thermal paste will fix it as it could just not be making proper contact with the inner heatsink.