They may be a canine that you seem to have an aversion to killing, but if there are as many as you say, it will probably take mainly trapping to cut the numbers down with maybe some gunning as a supplement. Coyotes have a huge territory and there can be a lot more in an area than people realize since they are smart and hard to see most of the time. Get on it as with that many around all the animals will be fodder for them, especially newborn fawns in the Spring. Contrary to what a lot of people believe they can also take down a full grown deer under the right conditions, one of which is a lot of snow with a crust on top that they can run on and a deer breaks through. It's not pretty when they take one down either, so hopefully you'll get away from that "kill a canine" aversion you said you have.