I have similar symptoms. Well, my furnace does.
A repair tech, who's opinion I generally respect, also suggested the
thermostat and seemed to be very sure that it was probably a battery
issue when I said I couldn't remember the last time I changed them.
The original thermostat was a round dial, nothing more than a mercury
switch on a bi-metallic coil, so I emulated that by taking the modern
digital set-back stat out of the equation and jumpering the two wires
together, which didn't change the symptoms.
BTW, my furnace is about 35 years old - Amana GH-120DE and I can't
find an induced draft motor, just a direct drive squirrel cage blower
at the bottom.
I replaced the fan and limit control, the flue is clear (enough).

Any help appreciated, but I can't afford to chuck it out.