Could it be a bad thermostat: I seriously doubt it, Unless the T-Stat has a setting to run the furnace fan and is "Stuck" and I do not think such a setting exists.
Could it be a bad control board: Odds very good.
Where is it located? If you go outside, and remove the 4 screws securing the cover for the furnace, then the cover you will see the board on the far left, A "Wing nut" holds a plastic backer in place, loosen (or remove) the nut and the board, and it's backer (Mounting bracket) should be easily removed part way, alas only part way.
You will see two heavy wires hooked to the board, ONE of them is the fan (The other is 12 volt battery power)
There is also another heavy wire hooked to the top of a black "Tower" or coil, that's the ignition/flame sense wire
And a molex connector with several wires,
Unplug all, have board tested by an Attwood service center (Perhaps Camping world) or...
I replaced mine with a Dinosaur board.
The dino board I used cost me 108 dollars shipping and handling included. The look and feel of the board was way better than the OEM.. ONe diffence is the board had some brass "Feet" and it appeared to me it would not hang onto the plastic backer/bracket properly so I added a pair of small zip ties to hold it (Bread ties would work as well). I tend to gush when talking about the dino board.. VERY WELL BUILT and superior in design too to the OEM. What I THINK caused my failure,,, Well.. On the Dino board there is an extra part that is not on the OEM.. appears to me to be there for the expressed purpose of making sure what I think happened to the original. DOES NOT HAPPEN to the replacement.. This.. I really respect.