Ariens snow blower won't start after rain.

Hi all, I have a 20 year old Ariens snow blower with the 10 horse power Tecumseh. When ever it rains and then gets cold I have difficult time starting the engine. It's easy to pull and get the crank spinning by pulling the cord and I can smell fuel. I replaced the tubing for the primer about 4 yrs ago. This has happened for the last two years. Once it starts it runs fine. A week later still starts up fine even when below 25 degrees. Once there's a heavy rain again, I have to heat the thing up in order to start. In spring I'm going to do some long over due engine maintenance. I'm wondering what should I do to address the starting issue. I was thinking of the carb, coil and plug. Electric start doesn't work and the pinion is broken again. I think the starter needs to be cleaned out and I'm wondering if there some kind of lube I can use for it. I will bench test it to see what's happening.
 
Any suggestions would be welcome and appreciated.
 
Thanks
Rob