Sounds like a faulty T-Stat.
I will describe one T-Stat or two that I have seen or tore down
I just put an "open Switch" type T-Stat on my Furnace (long story omittee) when the thing opens a small magnet holds the switch open so it take quite a bit of "PULL" to close it,, likewise other things help hold it closed when it's closed so more pull is needed to open it.
This causes the furnace to run "For a while" not "blink" on and off.
Another type uses a bi-stable bi-metal spring bar, it "Snaps" open and closed and again there is a "Gap" between open and closing temp's
That's the kind Atwood uses.. Practically they both work the same but the magnet one is easier to see (it is the 20 dollar one from ACE if you'd care to take a look)
Yours has lost that "Gap" is my guess so instead of waiting say 5 Degrees to turn back on it's waiting 0.5 Degrees.
