Hi Kit,
I myself have not worked on one of these older fridges, maybe some of our other RV tech's and fridge fixer folks have and can guide you better.
I have found this. From what you are describing, your thermostat either has internal issues, or the rod connection the to the T stat is broken somehow letting the rod to spin.
Since you can get the fridge to run on max cooling, in both gas and electric, the good news you cooling coil is still good and the gas burner and electric elements is still working. That is a big deal and in my mind means the fridge can be saved.
These older fridges had a combo T stat, gas control and electric element control running off a capillary tube style T stat that runs the whole temperature regulation for both, gas and electric. It's a simple mechanical setup, it my be the T stat not longer works right.
To fix this, ideally you can find an RV tech or a home LP gas appliance repair friends who can help you. The T stat is made and not and super easy thing to replace, but it doable.
This is what I found which may help.
On our site, in the files section are there two Dometic files
An exploded view of the parts of the system to help show how the parts fit together.
This one shows the exploded view of the RM 2500 series fridges
Dometic Gas/Electric Refrigerators Installation
https://www.sunlineclub.com/forums/d...o=file&id=5561
This one is the service manual that includes the RM 2510 fridge. I just uploaded this tonight for you.
https://www.sunlineclub.com/forums/d...o=file&id=5642
This is a thermostat I "think" fits yours. Confirm yours looks like this if you decide to order it.
One on Amazon, sold by Amazon, they only have 1 left
https://www.amazon.com/M-C-Enterprz-...UNO/ref=sr_1_1
And 10 of them in stock at Boat & Rv Accessories here in OH near Dayton. I have bought many things from them and have had good transactions.
https://www.boatandrvaccessories.com...mostat-control
OR, search on line for the MC Enterprises 2007199009MC Refrigerator Thermostat Control.
And here is a U tube I found showing the process. I'm not recommending you follow how he did it, but it shows what is involved. I would of pulled the Fridge out of the wall cavity and changed the T stat out in the open so I can get to everything easier. And had less risk of not getting the control rods hooked up right and maybe an easier way to get the capillary tube in place.
That said, it is helpful and he did a good job on the video. Seeing this may help you decide to find someone to help you with this.