Your spot on, Simply adding a stabilizer to the fuel preserves the fuel over a period of a few months.
Personally i do the following, run the engine without the fuel line attached therefore using up all the fuel in the carbs etc.
Then use any left over fuel in the boat tanks for the car. No need then to stabilize the fuel at all then.
However if you have a premix two stroke (fuel and oil already mixed) then you will need to add a stabilizer to the tanks.
Of course some do it otherways, this is just how i do it.
In all seriousness, a cold winter day with full sun on calm waters is brilliant. Just would call it quits if ice began to form on me......