Take a look at this. It is kind of a generic way to test any air conditioner.
https://itstillruns.com/test-rv-air-...r-7823972.html
It sounds complicated, but it is not. But it assumes you have a digital voltmeter with a capacitance scale.
Before you start, try to determine the make and model of your unit. When you find it, go online and download the manual. It will have good info that is specific to your unit. If you have the blue looseleaf notebook book that was originally supplied with your TM, it will be identified in there.
Next - assuming the temperature inside the TM is higher than 65 or 70 degrees, so the air conditioner will recognize that it needs to do some cooling ---
Turn the thermostat to COLDEST. Turn the operation knob to FAN. If it sounds like it's running as soon as you turn it on, you are hearing the fan. If that sound continues for more than 15-30 seconds, the fan and its capacitor are good.
Once the fan is running, turn the operation knob to LOW COOL. Very shortly, you should hear a deep rumbling sound. That it the compressor. It should keep running, and you should get cool air out of the vent.
If one or the other of these tests fails, you need to get up on the roof of the TM. Before you put weight on the roof, bring up a couple pieces of something stiff to distribute your weight. A couple 2x2 foot or 3x3 foot pieces of plywood or masonite or other thin board is ideal.
Unplug your TM from all shore power! Get up on the roof and take off the fiberglass shroud. It is probably held on with 3 or 4 screws, easily visible from the top. Don't drop it - it will break, and it is expensive. Inside, you will probably find a small (6x6" inch?) metal box with wires inside. Open the box. You will find a couple more-or-less tubular devices inside. These are the capacitors. Do either of them show signs of destruction? Burst open? Liquid leaked out? Smoke damage? Soot on the wall of the box? If so, you know you need to replace that unit. If not, proceed to the digital voltmeter tests.
Get back to us when you have identified the make/model of the air conditioner, or when you have done the checks and have some results.
Bill