Cool...that sounds like a super easy temporary fix and sounds like the best permanent is as easy as a thermostat swap out. Many RV folks do a thermostat upgrade as a routine "mod" anyway. The stock thermostats are none too accurate and there are many "upgrade" instructions floating around to change up to a Hunter Digital thermostat available at any Walmart for under $20
Thanks for posting a follow up "what works" on your problem.
Many times folks will post a problem, but, no follow up on what the outcome of the issue was and what the "fix" was...not much help for those that follow.
I'll file that away in my "what works" fix it file!