Silas,
You show great care and initiative, and that will serve you well.
Now, the problem with the advised monthly "exercising" the generator is that this old story is perpetuated by the people that sell the parts for said generators. It is bunk.
The varnish that was used on windings that had moisture trouble was replaced during WWII with the new material that does not absorb moisture at all.... So, there is not reason to warm those windings at all during storage.
The old fuel gumming up the carburetor is a real, but easily solved issue. In spite of you might have read, old leaded fuel was worse that the new alcohol loaded fuel for gum, but the new alcohol fuels can become corrosive. No good choice here at all ever. The real answer is to get the fuel out of the carburetor. If the generator has an electric fuel pump (many do) then disconnect that pump with the machine running and let it run dry. If it does not, find a place that you can disconnect or clamp the fuel line closed. It is even better if you can find the carburetor (I recently saw a new Onan and it took a while for me to find it) and you can get at it to actually drain the float bowl. That is what you really want to do.
I have several generators, I don't "exercise" any of them. One of the Onans will be 50yo soon and the is a Honda that is a couple years newer in the collection. All are drained and left alone - sometimes for a year or more and I know all will start as soon as they have fuel and a reason to start. That reason being that I pushed the button or cranked it up.
Matt - Long time engine professional