I don't have any data to prove it other than shooting myself but yes soggy feathers affect accuracy especially with fixed blade broadheads where many times the control of that raised helical is keeping things going straight. That and they just look like crap.
My buddy Frank treats his with the stuff made for Whisker Bisquit rests by CAP called "No Snow" He gives them 2 or 3 coats ahead of time and these puppies are as waterproof as you are going to get feathers. I've seen some of his arrows that have passed through animals and the feathers remain bone dry, and even shed blood.
Combined with a Cat-quiver protecting them, they remain pretty darn weather proof.
Good stuff.
I personally have almost zero patience for feathers even though I do like them. I keep switching back and forth but am starting to just lean entirely to vanes. I worry too much about downpours and stuff even though I know I could waterproof them nicely, and wind up hunting with a vaned arrow anyway on a rainy day. I shoot pretty much nothing but mech. broadheads so I can get away with light small vanes that carry more downrange trajectory and energy even though they weigh a little more than feathers........less drag.