Jim,
John Wayne is the closest to a good response so far, but he missed one thing. Run the carburetor dry, that is how you keep it from getting fouled with old fuel. Stabil is good, but the fuel can still dry out in the carburetor and leave stuff behind. Running it dry can mitigate this. There are two ways: (if you don't want to put in a valve)
First is if it has an electric fuel pump (does it have a prime switch?), find it and while the machine is running, disconnect it.
Second is the find where there is rubber line that you can pinch off (this is not good for the hose, but it works).
Now, in any case, get some "Fogging Oil" and fog the engine (spray oil into the air intake) the as it is running out of fuel. When it finaly studders to a halt, close it up and leave it alone until you need it again. It will start with some smoke, but that will last only a few moments.
Now is where I tell you that the start it regularly is put forward by the people that sell you parts. The idea of running it at load to keep the windings dry is left over from pre-WWII. The varnish used in such windings in the last 70 years has not been a kind that absorbs moisture.
Matt