Digital to Analog Thermostat?

This may not apply because of the action WRT the air conditioner.

Our 1991 furnace was giving that sort of trouble on an intermittent basis when we first got the rig a year ago. Had to keep turning it on and off with the thermostat to get it to get going again.

After enough of that I had the thing apart and discovered the burner with the furnace was rusted out where there is a sort of cover over the burner with many little holes in it--big parts of that just gone! So the flame was really wonky.

anyway got that fixed and no more issues like the furnace quits cycling and all that as described in the OP.

Before getting that far, I had also replaced the old analogue thermostat with a new digital one. That does work great, but was not the actual problem as it turned out.

However, the air conditioner business in the OP may be a red herring to the furnace problem or else truly related to it, don't know.

The problem may be something other than the thermostat.

To the FURNACE the thermoat is norhing more than a switch ON/OFF that's all, in fact I have a mechanical T-Stat and it is just that a SWITCH. nothing more. The switch has a magnet on it and a bi-metal (means two different metals) spring. as the temperature changes. the spring either winds or unwinds due to the different expansion of the metals. when enough tension is on the connection SNAP it opens. and likewise as the RV cools. it closes.

Your Electronics may use a solid state device or a physical relay to operate the same way.

THe furnace however has a comptuer that controls it. The computer looks to see if the T-Stat is OPEN or CLOSED, that's all it sees. OPEN/CLosed if it is OPEN the computer sits there. when it CLOSES the computer does a bunch of things in order Note the "Loop" part

Starts the blowers (There are two with a common motor)
When the sail switch closes (Air flow is up to specification)
Set fail counter to ZERO

(LOOP Start)

If fail counter = 3 Go to FAIL
it opens the gas valve and starts the ignition sparking
It then looks to see if the thermocouple is sending 0.480 volts back to the control board. If so exit loop.
Turn off gas.. Wait for purge. increment FAIL counter (+1)
Return to LOOP
If flame is sensed wait for a T-Stat to open
If over temp sensor (T-Stat) set FAIL flag
Wait
Turn off blower
If T-Stat was what cut off Return to Top and wait for T-Stat to close
If Over temp cut off go to FAIL
FAIL. Set FAULT light and flash Ignition Lockout on diagnostic LED.
Wait for THERMOSTAT to open or power to be removed.

When the dealer turned the T=Stat off and one.. he reset the FAULT indicator on the control computer.

Possible causes includ
Bad burner (Dirty/clogged)
Excess spark gap (not sparking)
bad thermocouple (Rare but possible)
And my favorite Bad control board.

My Atwood has a "Single wire" ignition system. the spark point IS the thermocouple and one wire carries both about 1,000 volts to light my fire and less than 1/2 the other way to confirm it's lit.. Think about what happens to teh sensor chip if the ignition kilovolt hits it (Pop goes the sensor).

My Dinosaur board has a nifty little protection device right where I think there shoudl be one to prevent... Just that.

No. At least not without also replacing the control box up in the lower unit and possibly pulling some additional control wires. I converted mine to digital from analog.

I like my Dometic single zone CD thermostat and have not had the issue of not continuing to work overnight. I don't use it in storage.

I would ask for an additional warranty replacement first. This is new technology for Dometic and I think some early ones had issues. Your dealer may have had an older one on the shelf.