I've never had a swallowing/breathing problem before, but when I try to WILD I keep my mouth closed at all times, and I lay on my stomach, so I don't know if that has something to do with me not having those kinds of problems.
I breathe through my nose, also, and I usually get into the first phases of SP, so I don't think you need to breathe through your mouth during WILD's, in order to have SP.