Welcome to your iodine experience :) Other people have experienced swollen salivary glands. The glands concentrate iodine and will take advantage of all that iodine passing right by :) It won't last, but I don't know how long before the swelling response dies down. If I was that uncomfortable, I'd take a break, take some more, take a break, take some more, till they're finally satisfied and you can get on with the rest of the program. I only had a mild version of it; I don't know if made a difference that I took Iodoral, not Lugol's. The iodine will still end up in the salivary glands, no matter the path.
Excellent results with your painting! Yes, Lugol's can be used vaginally. The experiences with it have been hit or miss. I remember somebody saying they thought a q-tip soaked up too much of the iodine for painting. This thread has some ideas; you might search for others.