I don't have a generator for my TT, but the 4,000w gen I have for the house (contractor style, LOUD!! lol) I store empty for long periods. After I use the generator, I put Sta-bil in the fuel tank and run it for 20 minutes or so to get the additive throughout the carb. Then I shut off the fuel and run the gen until it empties the carb. Then I drain the fuel tank empty (into another container). If it was run for multiple days, I'll also change the oil.
Then I put the gen in my shed. There it sits. When I need it (last two hurricanes), I fill up the fuel tank, check the oil (ALWAYS check oil when filling fuel tank!), and start it up. So far this has worked great for me - last 10 years or so.
This is for long term storage. If I knew I would be using the gen within the next month or so, I would keep it filled and start it at least once a month. If I had a gen for my camper (as most gens in this thread are), I would be starting it regularly to make sure it runs.