Those Dometic thermostats are not very good and I never could get one to work right in my unit. And I know many others that have the same issues. Unfortunately they are not replaceable with a standard thermostat since they communicate digitally to the control board in the AC unit. There's a workaround where you use the Dometic as an A/D converter and use a standard thermostat. I tried that and it works but it's complicated.
I eventually went with the EasyTouch RV thermostat and it's much nicer and works great. A little pricey but they are the only ones out there that have a replacement product for the Dometic.
As the EasyTouch guy told me they are thankful every day for Dometics lousy engineering. It's what keeps them in business!