Hi and welcome.
The RM661 is a fairly old fridge. I think it is a mid 80's vintage. These older fridges had to be very level to work correctly and only run that way. While they might run for a while off level, they can overheat easier. The overheating can lead to blockages in the cooling unit or a cooling coil failure due to a leak. Since you just bought it, any of that long term damage to the cooling unit was before your time.
What you are describing by the electric element area being warm but not getting any cooling points to a possible cooling unit blockage or failure.
First off, have you leveled the camper front to back and left to right that if you put a small level on the fridge shelving or on the floor by the fridge, is it level in both directions? If not, shut the fridge down and level it.
The issue is the overheating part. If there is a blockage or a leak in the cooling unit, all the heat being put in by a gas burner or an electric element makes the overheating situation worse. And if left on long enough, the steal can get so hot it may become a fire hazard. In this case the fluid inside the cooling unit is not flowing keeping the steel cooler to not get so hot it overheats and creates issues.
You can try feeling the pipes in the lower part of the fridge, but not directly next to the boiler/heating area. They should be very warm if the burner area is also warm. If they are ambient air temperature or only slightly warm, but the boiler area is warm, and the fridge has been on for several hours, it can point to a blockage in the cooling unit or a cooling unit failure.
If you see yellow residue at the bottom of the gas burner area or smell ammonia, that also is a sign of a cracked cooling unit and all the fluid leaked out.
If you do not see any yellow residue or do not smell any ammonia, that kind of suggests it just blocked in the tubes. Some have tried to correct this by taking the fridge out and rocking it around and upside down trying to break free the clog/blockage. It's a 50/50 shot. Sometimes they get lucky and the clog breaks free, other times it jams harder or won't budge loose. I have never done this but we had a recent member do it and he heard the clog let go as fluid started running in the tubes. It is a trial and error approach and a last ditch effort to get it to work, IF the unit is blocked/clogged.
Hope this helps
PS, they do sell replacement or repaired cooling units. A remanufactured one for a RM661 is $415 plus $65 freight from RV cool. These guys http://rvcool.com/index.htm
Here is the remanufactured units pricing
All units purchased using online payment will ship within a few business days
. I have bought an all new unit from them, a much newer model, and was pleased with the transaction and the quality. They do not seem to offer a RM661 new coil.
It seems Amazon is selling and all new one for $642 including freight. https://www.amazon.com/RM2500-RM2501.../dp/B011AF4RZC
Suggest before buying from anyone, call them and confirm the model and serial number of your fridge that what they sell you will fit it.