Never had a fur problem on my venison. I keep the skin on the carcass till I get home. Then I hang it and skin it at home in a utility tent that I keep around just for that. Keeps the wind out and if its REALLY cold I can put a propane heater in there just to keep my hands from going numb.
I hang the carcass from a tripod I made out of scrap angle irons and chain. Then I skin with a very sharp skinner that my dad made. Its not what most modern hunters would consider a skinner (the blade shape is more like that of a pig sticker) but it works well for me and I've been using that knife for decades to do the job. Skinning is quick and easy and I can have the whole carcass skinned and quartered for butchering in about 20 min by myself. Usually it takes more time if there's someone else "helping..." Legs come off at the knees - you don't need a saw or other equipment to take them off. Then cut everything off from the bottom up. I hang my carcasses by the head so I work with the hind quarters first. I should clarify that when I say quartered it means I take the hind legs off then the front legs then the lower half of the carcass and then finally I'll decapitate the carcass and its off to the processing table in the kitchen. I do all this by hand with that one knife I mentioned.
My neighbor and others I've spoken to have mentioned using a propane torch to get rid of the fur that gets in your meat. Quick and easy they say, no big deal.
This is who we are, what we do.