The replacement model is a rm2351 or 2354.
Either the thermosat works or dont, you can take the tip of the thermostat and set it in a cup of ice water to se if it clicks if it does its ok.
Check the back of the selector switch knobs for being worn out/not turning the switch.
Yes the flame flucuates somewhat but not much.
If you hear a bubbling sound in the back when on flame the cooling unit is clogged/bad.
While running a seperate lp appliance the lp water colum to the fridge should be 8", if not its the lp regulator a lp drop test wil tell you that.
The flame should remain constant. The thermastate turns the burner on and off but does not control the flame. When it gets cold enough it shuts off. If the flame is to small that could be the problem. The stack and the burner tube should be cleaned. That might just take care of it. Does it work OK on electric?
The flame will stay the same height untill the refer reaches operating temp. About 36 deg F is optimum. When it reaches the corect temperature depending on wheather it is an auto energy select or not it will either shut down or the flame will get smaller. And Kali.... The LP should never go below 10.5 in. W.C. the regulators are 180,000btu.