I'd recommend running it until its dry. Make sure that when it starts
to sputter and die that you move it to full choke to get as much gas
out of the carb's bowl as possible.

Depending on how crazy you are about your tools (I am) I drop the bowl
to clean it out (dirt and extra gas left over) reassemble it. If the
gas tank is a steel one, then I spray a light coat of oil into the gas
tank to coat the walls (prevents rust). You can even coat the carb
with
the same spray to prevent rust damage over the years.

If it were long term storage i'd recommend changing the oil if its a 4
cycle and
even throwing some oil into the cylinder to coat the walls (fog it like
a boat engine)

You should also remember to lube up the snowblower as per your owners
manual.
Some require a little move care than others depending on if its a 1 or
2 stage unit.
Single stage units should have their auger belt checked to avoid that
breaking when
it comes to the snow season.

With a bit of maintence these things can last a lifetime.