I did this exact mod in the spring. Although I have a Coleman AC unit. I had to buy a new control box and a new thermostat.
The new control box went into the space left by removing the old control box ( your current one with the knobs on it).
Move AC power from old control box to new control box.
Move air conditioner control plug from old control box to new control box.
Run a 4 conductor wire from new thermostat to new control box, (Unless you have one already run from factory, mine didn't)
Use existing wire to furnace and connect to new thermostat.
Connect compressor, fan hi, fan low and +12 volt between new thermostat and new control box.
Supply +12 volts and ground to new thermostat.
As mentioned I have a Coleman AC unit , not sure if the same plug and play parts exist for a Dometic unit.
Here is a link to my post when finished