I am not a profesional rv service person, But I have some ideas for you from my service book. From what you indicate, it points to a problem with the thermostat, the knob you are turning to try to regulate the temp. It could be a faulty thermostat, an incorrectly locatedend of the thermostat capillary tube , an incorrect size of the bypass screw, or dirt in the thermostat valve.
The good thing is that from, your description , the cooling unit itself is ok, because you say it works fine on electric.
There are still parts available for the RM 100. Do an internet search.
Here is a # for Dometic technical service 219-463-4858.