12 volt may come from A/C or Furnace depending on how it is wired.. Most home T-Stats default to Furnace (But may switch to A/C if COOLING is selected) If I were wiring one for a motor home or travel trailer I'd do it that way (Default to Furnace) I might even not bother with selecting A/C as the source when COOL selected.
I would hop outside with your volt meter (or test light but in this case the meter MIGHT be better)
On the furnace control board you shoudl find a multi-pin (MOLEX) conector, TWO of the wires will be BLUE. See if they show 12 volts across them.
Also check for 12 volts from EACH of the blue wires to ground (only one should be hot)
On many furnaces on top of the blower housing is a switch, it is also a circuit breaker, cycle it and see if you now have 12 volts.
Now the good news, or bad depending on point of view.
I am not sure, bit I do not think the control board supplies the 12 volt.. That said, there may be a TRACE on the control board involved but no electronics.. Just the trace.
Atwood control boards...... When mine failed (Different sub-circuit) and I pulled it and replaced it with a Dinosaur board I was impressed by both of them.. In the case of the Dino board. by the high quality of construciton and a few design features.
In the case of the Atwood board.. The impression was well... Let's just say it was the quality there that impressed me as well.. and it was NOT a good impression, epically side by side.
I do not recall if there is a fuse on that board or not, but if there is.. no there is not, the circuit breaker takes care of that.
After that it gets tougher.