We have replaced the filters and the limit control. Didn't help.

If the blower continues to run with no burner and the space temp is not satisfied (still calling for heat) then there must be an issue with the fuel delivery systems such as the pilot light goes out. If the pilot light flame is extinguished the gas valve will shut down. Could mean a defective thermal couple ($20 part).

If the blower continues to run with no burner and the space temp is satisfied, then you may have set the thermostat to "FAN ON".

I agree with Ivan and think checking to see if your stat is set for fan ON or fan AUTO. Come back and tell which it is before doing anything else

Our thermostat isn't that clever - it doesn't have the modern options, only temp. It has an automatic switch, no pilot light. Keep the ideas coming!

Is there something in the furnace that a new stat would trigger that might have been inadvertently switched to FAN ON?

No little switch marked ON/AUTO frequently on the bottom edge. very unusual. Any chance of a pic or brand/model