Not even my lawn mowers want unstabilized gasoline more than 30 days old. Stabilizer goes in the 5-gallon portable tank I use right at the pump so they all get good fuel all year long, not merely in winter storage. (Still mowing here in Virginia.) The pre-packaged 2-cycle gasoline used in my leaf blower and week wacker are factory stabilized.
I typically run all gasoline engines out of fuel (by closing the fuel tap) when I'm done using them and leave the stabilized fuel in the tank. I have a small roadster that is rarely driven in cold weather and Honda specifies to run the engine monthly until it reaches operating temperature which they define as two fan cycles. Makes sense for other powered vehicles.
Full fuel tank of stabilized fuel should be fine over the winter. Run the engine 10 minutes to run the stabilized thru the entire system.