Full disclosure:  This is not my idea.  I liberated it from another hunting forum, over where they use bullets and stuff.  But I feel pretty safe assuming that most people hard-core enough to take to the hills with a primitive bow are probably processing there own meat
I might be really inept at field dressing, but my meat always ends up with random hairs on it.  Sometimes quite a bit.  This year's was cleaner than usual, since I actually killed it in a level spot for once in my life (I have a gift for running into deer on the sides of horrendous, high desert canyons miles from the nearest road).  But there was still some, and it is a pain to remove.  At least, it used to be.
Here's what you do:  Before you start the clean-up-and-wrap-up phase of processing your critter, take a half cup or so of wheat flour.  Add water a few drops at a time and knead until you have a blob of nice, sticky dough:  Not wet and slimy, but sticky enough that it adheres to your fingers just a little bit.  Keep it in a cup next to your working area.  When you find a hair, squish that blob of dough onto it, and more than likely it will pick it right up off the meat.   If it doesn't, use a gentle wiping motion and it almost certainly will.  Squish the dough a few times so the hair gets enveloped and doesn't go right back onto the meat.
If the dough dries out, add a couple drops of water and knead it until it's sticky again.  If it gets hairy, knead it a little.  After a while (about half a deer's worth, for me) it starts getting pretty slimy and quite working.  So I just mixed up a new blob.
Easy!  I sure wish someone had told me about this 30 years ago...I could have taken a second career with all the time it would have saved.  Or practiced shooting more.  Or something.  Anyway, there you go.  Give it a shot and see if it works as well for you.
Thomas