Nut, a lot depends upon the time of the year when I'm hunting. Let's start with right now. It's hotter than blazes and I'll try and sneak out if I can to call for the camera. I do carry a rifle in case a mangey coyote should come in. Since less than 1% of coyotes that contract mange survive the infestation, I regard this as doing him and all the other coyotes in the area a big favor. Well furred coyotes are allowed to go on about their business. Even if I'm on an ADC project, summer pelts are worthless for anything, including tanning since the hide hasn't thickened enough to anchor the hair.
Fall and winter are fur months. I shoot to kill. I haul my take home and skin them in the comfort of my fur shed. The pelts are frozen for washing/ fleshing/ stretching and drying later. As a taxidermist, I often skin some of them for lifesize mounts or pedestal mounts. My carcasses are disposed of either at the lanfill, or I have a few remote tracts of CRP grass which the owner has posted "No Hunting" and allows me to drag the carcasses out to the middle for the vultures, crows, and possums to feed on. Check with your local Conservation Officer or county landfill ppl to see what the law allows. Either way, do not just dump them over a bridge or in a roadside ditch. I can't count the number of times the local Sheriff has asked me to respond and identify what was reported to them as a satanic ritual dog sacrifice. If you have a furbuyer or trapper in the area you know of, ask them. If you don't want to skin your kill, find a young trapper or hunter that would like them. When I used to trap, I hated dealing with possums, but they were a necessary evil. I found a couple young junior high-aged kids that were just starting out and gave them a half-dozen possums a day. At fifty cents a piece come auction time, they were a lot more excited at the prospect of that extra money than I was about the amount of time and mess I had to deal with. Coyotes on the hoof would be a treasure for someone that age.