Step one may be to get a clear picture in mind of what a thermostat really is when we ask it to do something.
Basic point is that the thermostat is just a set of switches that turn on other parts of the RV. We can then look at it as a simple set of switches with switches on it that tells it what we want to control.
It sounds like you made a good start by looking for info on the particular thermostat you have, so it may be time to just dive deeper into that info you have.
I might first want to define what portion of the thermostat is not working?
I has several functions, so just saying it is not working, really doesn't define the problem close enough. What functions are working and which are not?
When you turn it to fan only, does the fan work?
Turned to cool, does the air seem to work?
How about any of the various heat sources work or not?
When you get a better idea of what does or doesn't work, you can often find info in the manuals which lead to knowing which wire does each operation. There will be wires going from the thermostat to the AC, wires going to the furnace sources, etc.
One of the bigger problems with thermostats is that they fail because the inputs and outputs fail!
If the wire between the thermostat and the furnace comes loose, it looks exactly like the thermostat failed!
One way that may lead to info on which wire does which function may be as simple as pulling/ tugging the set of wires attached to the thermostat body, up out of the wall to get to the wires added by Winnebago. Tug and wiggle somewhat gently if they are tight but try to find the connections which are often 8-10 inches away from the thermostat.
If you get those Winn. wires exposed, look for wire ID stamped on the wire ends!
If finding Id, then go to this chart to decode that label ID:
https://www.winnebago.com/Files/File...ical_guide.pdf
You may find ID like these? Click this snip for better view!
This is just part of the thermostat wire ID listed (maybe 150?) so you will have to find the correct listing for what you have but the idea is that you will then know which wires to test, based on what has failed!
In this example, check the simple stuff like 12volts and ground getting to the thermostat as a first step?
If the heat is the problem, does wire JJ for the heat get where it should? Maybe that connector in the wall has fallen off? Or maybe it is corroded where it connects at the furnace?
I would not change the thermostat until I checked the wire nuts!
If we are at home and the TV stops, first check that the neighborhood still has power!