If running on part-choke helps, I'd do that to get the task at hand done until warmer weather hits and you can look into it further. Especially for an engine that doesn't like the cold anyway.
I have also found that with engines that have sat, that gum and varnish from old gas will sometimes redissolve a bit in newer gas, and if you can get them to run at all you can sometimes improve them by simply working them and running more fuel through them. It applies solvent (new gas) and suction (engine vacuum). Not a long term fix, but sometimes it is enough to get you through a job.
I do agree that any type of enclosure to keep engine warmed air closer to the engine will likely help, with or without moisture involved.
