You might also look for white stain on the large round part of the cooling rods. It's about the size of a liter bottle if it's really hot to touch it's toast. There is not a great deal of mechanical stuff with these units. They tend to work or not. Consider residential fridge or new unit if that's the case. You might find that repair and replace are close in price. This can be dyi project, it's time consuming be doable with few tools.
May or may not be a "diy" fix. It strictly depends on the refrigerators needs, once it can be properly troubleshot by an expert. It may also need a new heating element. A new refer can run up to $1,500.00, so, there's likely a fix that can be done for far less than $1,000.00 in all likelihood. All refrigerators take two men to lift them out of their cavity and then tackle dismantling the cooling tower from the refrigerator, then either taking it in to the shop for the repair patching, or, sending it off FedEx to the refrigerator repair shop that can handle a proper fix. Look to spend about $300.00 doing this if you have to send it via FedEx roundtrip. You've also got to be trained and knowledgeable in re-caulking the cooling tower to the backside of the refrigerator - then having two men lift the refrigerator back into it's cavity onboard the RV. It's likely impossible for one man, especially if he's a beginner at this type of work.