A typical thermostat includes an adjustable "delay" (anticipator). The delay enables the system to cycle less frequently. So the furnace will wait before firing until the temp is a bit below the desired temp, and then it will run longer raising the temp above the desired temp. This gives the furnace (or AC) time to rest between cycles.
In my PUP, the delay was set to an extreme, and we had the same issue...freeze or roast. My simple mechanical thermostat was easy to adjust. I pulled the cover and reduced the delay. Much better, and the furnace still does not cycle on/off too frequently. (In a PUP in cold weather, this has the side benefit of not attempting to raise the temp from about 65 degrees to about 75 degrees...a huge swing in what amounts to a "tent".)
Here's a video for a mechanical thermostat:
For digital thermostats, it's likely to be a programmable feature:
So, it definitely makes sense to isolate the thermostat from any undesirable influences from exterior walls, drafts, metal that conducts outside heat/cold and so on. But you described a situation where you're either freezing or roasting, so I suspect adjusting the delay will solve that problem.
P.S. It's not surprising that your 98 degree breath affects the thermostat. Also on an interior surface, the surfaces should be about the same temp as the air. Things tend to stabilize, with cabinet faces and partitions absorbing heat from the air and achieving essentially the same temp as the air....just as in your home. In my PUP, the thermostat is on the front face of a kitchen cabinet...isolated from the outside wall by the depth of the cabinet. I presume your thermostat is situated somewhere well away from an exterior wall, so the temp of the mounting surface should not affect the thermostat in some bizarre way. But you could isolate the thermostat from its mounting surface with some 1/4" closed cell foam:
https://www.amazon.com/dp/B001G8E114...a-309563662940
That was only an illustration. You should be able to find something 6" x 6" x 1/4" or so at a hardware store.
__________________
Jim & Renee
2020 Jayco Jay Feather X-213
previously 2014 Forest River/Rockwood HW 277
2006 Ram 1500 4WD Crew with Firestone Airbags
Every weekend boondocking in the National Forests or at
Lake Vallecito.