Figure out how fast your arrows are travelling. Let's say 150 fps, just a round number to work with. Ok, and your deer is at 20 yards, that's 60 ft.
60/150 = 4/10ths of a second.
That's how long the arrow is in flight between leaving the string and hitting the target. Actually, since the arrow is slowing due to the effect of drag on the surface area, a bit more, but 4/10ths is a pretty easy number to work with.
By the time the deer has heard the sound, processed the sound, and has fired off messages to the muscles, and the muscles have begun to respond, the arrow has already arrived. Almost every video I have ever seen of a deer "jumping the string" has the deer moving before the release. In one, the deer actually drops AFTER the arrow pass!
The real trick is not letting the deer see you drawing! Or pick dumb deer. I prefer dumb deer.
Go thru your arrows one at a time. Pull on each of the feathers, front, middle, back. I always seem to find areas where the glue loosened up somewhere on a fletch and is the buzz culprit. If they are all nice and tight, then try rounding off the backs of the fletching, or even cut them down lower.
As for target arrows, often I will leave the very back of the fletch high on purpose to make them do that lovely "Robin of Loxely" sound that the movie sound people are so fond of. Now if I can just figure out how to make my bows do that cool leather creaking noise. So far, the only sound I can make them do is the dreaded "tick....tick...BLAMMO-BABY!"