I replaced the POS Coleman thermostat with a Honeywell a couple of years ago. The Honeywell does require a battery as it’s a residential thermostat that normally wants power from a transformer. In an RV you don’t have that power so you need the battery. Nice thing about the Honeywell is that there is a back lit display so you can see the setting when DW decides its too cold/too hot at 3 AM.
The Love Your RV post details the swap process, including the external hi/lo fan switch
https://www.loveyourrv.com/rv-thermo...focuspro-5000/
I used a more attractive switch
https://www.amazon.com/gp/product/B0...?ie=UTF8&psc=1