Not a stupid question at all. There is no need to put additional stabilizer in the mower, if the gas already has stabilizer in it.
If your gas does not have stabilizer, you should either:
1. run the carburetor dry and drain the fuel in the tank, or
2. put stabilizer in the gas, fill the tank and run the mower for a while to get the treated gas into the carburetor. Then you can just leave it that way over the winter. Shut off the fuel valve if it has one, but don't worry about running the carburetor dry, it's not necessary.
I use Stabil Storage when I fill my gas cans, and I never worry about running the carburetors dry or draining the tanks in the machines. It's better to fill the tanks in the equipment for storage, because it will prevent condensation in the tank, and that gas will be just fine over the winter. The machines always start first or second pull in the spring.
I take the gas I have on hand, and put it all into one or two cans, getting them as full as possible, and I leave the empty cans until next year to fill with fresh gas (which will get stabilizer when they are filled at the gas station).
I have 2x 5 gallon, 1x 2 gallon, and 1x 1 gallon gas cans. I usually end up with a 5 gallon and a 1 or 2 gallon that are full over the winter, which gets me going in the spring, and I do not have to run out and buy gas until the full 5 gallon is used up.