The original question was should the OP keep the fuel tank full, or should he wait until he needs to use the equipment to fill it. It's been my understanding for years that a full tank leaves very little room for moisture-laden air, and thus should be kept full. I use only ethanol-free gasoline in my small engines, chain saw, weed eater, leaf blowers, etc. My chain saw is 20 years old and gets used a lot in fall and winter, and I have never had an issue with it. Same with the weed eater, 22 years old, never a problem. Both are Stihl FWIW, and I use only Stihl synthetic oil. But, in both I fill the tanks after every use. This goes for summer or winter equipment. I've never had a fuel related problem with any of my engines. I'd run ethanol-free in my vehicles if I could afford it.
Another reason to always keep the tank full is simply have it ready to go when you need it. If the power fails, I don't want to have to take the time to fuel up the generator. I want it ready to start at a moment's notice.
Oil...I would never leave an engine without oil for any reason. While almost no one has access to my equipment but me, I don't want to have to make an admission sometime that I started an engine with no oil in it. IMO there is never a reason to leave an engine without oil.
Just as an anecdote, I had a Ford 3000 diesel tractor that blew the radiator in 2006. I had plenty of tractors and little time, so I didn't get around to fixing it for 16 years. When I retired I tackled that job. The engine had set for 16 years without moving, barely even touched. I changed the oil, put a new battery in it, and it started up the first try. Same diesel fuel that had been in it when I parked it. The fuel tank had been filled when I parked it. Take that for what it's worth.