did you slide the fan section out and wash out the fins on the squirrel cage...if there covered with dust and dirt the will not catch the air to get it thru the heat EX up into the space...yes on the fan cycles after the space satisfies jusr residual heat off the heat exchanger...keeping in mind the fan is the last thing ON when heat calls

and the last thing to go OFF when it satisfies

..and the fan cycle is thru the limit there not the stat the limit is cycling the fan reacting to leftover heat after the burner