[cabmandone]

If it's lighting but the fan isn't coming on, it's not the thermostat. If it's lighting and the flame stays lit but the blower doesn't come on, it's not the flame sensor. If you have a modern furnace with a circuit board, I'd say your fan relay on the board is going out. If you want to test you can turn your fan switch on the thermostat to the ON position. This should energize the fan relay and bring on the main blower.
As a note, if you ever question a thermostat failing, make a small jumper wire, remove the wires going to the R&W terminals on the circuit board and put the jumper on the R&W terminals. This will signal a constant call for heat and the furnace will run until you pull that jumper.

Edited due to incorrect information regarding limit switch.