I took another look at the snowblower today. I pulled the float and valve out. It was physically stuck in place and I had to yank it out with a pliers. It was froze shut. After cleaning it up I gave it another quick go to no avail (not too surprised).
Next I removed the carb from the engine. It seems pretty gummed up. The choke and throttle both turn with a decent amount of effort.
Right now, the whole thing is sitting in this carb cleaner bath. I've used it in the past with pretty good results. I usually let it sit overnight.