After dipping in egg batter, I give paper patches a light dusting with flour. Deep fried, they are almost as good as potato chips
Now my serious response; the paper patch serves the same purpose as a sabot or a cloth patch on a muzzle loaded round ball. Paper patches are supposed to separate from the bullet on exiting the muzzle. Egg/water solution acts like glue and patches may not reliably or completely separate from the bullet.
My best results have been with dry patches. Damp or wet, patches are easier to apply but again, may or may not separate from the bullet on firing (depending on the particular paper you are using).