I store my lawnmower in the shed with a full tank. My generators are stores in my attached garage with empty tanks.
My reasoning is that the shed has large temperature swings and the change in pressure in the tank will cause air to siphon in and out. If the tank is full, there is little room for air to move in and out.
The temperature in the garage is more stable, so not as much of a worry about humid air entering the tank and condensing inside the tank.
I also use Stabil Marine 360 in the gas and 1 oz of 2-cycle oil in a 5 gallon gas can. I burn this mix in all my OPE year-round and winterize my motorcycles with it. I'll slosh it around the metal fuel tanks before draining the tanks, then I'll run the carb dry. It leaves a thin film of oil inside the carb and tank to prevent corrosion and rust.
For extra piece of mind during long-term storage, you could get some engine fogging spray and spray it in the fuel tank as an extra rust preventative coating. You'll want to make sure the the tank is drained and dry inside so the fogging oil sticks to the metal.