I don't know what non-oxygenated gas is, but if it contains ethanol then that is the problem. Ethanol breaks down into water and really can gum up the carb. It is better to have used 100% gas in the generator before shutting it down for the season. Best yet to run the carb dry (out of gas) before storage. That is what people do with small engines all the time.
Regular 100% gas will gum up the carb also after a month or two.
When you say run the generator, there is two things going on.
1. the actual electricity generator is the electric motor looking "head", and this needs to be turned over monthly in an ideal situation. In reality, it can sit for years untouched and still come out working when restarted. Try to keep it covered and keep dust and debris from getting into it.
2. The gas engine, which is just the same as any small engine, lawn mower, chain saw, tiller, go cart, can sit for years safely if the gas is not in the carb.
It is also more ideal to start and let the gas engine run once per month. At idle is finr for 10-20 mins, but under load is what a lot of people suggest too.
Again, I have let these things sit all winter and had no issues come Spring. I just revived a friends Onan 4000 that sat, with gas in it, for 13 years, though the carb needed a serious cleaning, which I remedied by just buying a new carb ($40).. It runs and generates electricity just fine, though yes, I would prefer it had not sat so long. Not ideal, but not death either.
So, ideally, run the carb dry. You might need to disconnect the fuel line from the gas tank to quickly run it dry. Or alternatively, you can remove the "bowl" from under the carb and just let it all drain out.
The gas in the tank will need some gas treatment gas saver like Stabil to keep it fresh longer. This will not guarantee good gas come Spring, but will extend the life of gas if you cannot remove it for some reason.
