Sounds like symptoms are pointing to your carb.. If you're keeping it running with the choke, that doesn't sound like water in fuel.
I'd be checking the carb & intake for air leaks. You can do this by using a water spray bottle (full) & w/engine running, spray the connections at the carb to intake, & intake to head(s). Take your time & pay attention & only do one connection at a time, to rule each section out.
If it has an air leak, it will run extra crappy or even shut off, because of the water getting in..At the connection that causes this (if it does), that would be the leak to fix.