This is a long shot, but I read that people with MG (and other autoimmune diseases) sometimes develop Sjogren's syndrome, which causes dry eyes and a dry mouth. People with Sjogren's don't produce enough saliva, and they often have to take a sip of water to help them swallow each bite. Is it possible that's what's going on? Like I say, this is a long shot. Certainly MG on its own can cause swallowing problems when it weakens the muscles you use to swallow.

Abby