Foxpro is good to go, never had a primos.
If you're seeing 3-5 dogs in a day on a regular basis, then you are definately in a high coyote population area. Either that or you are simply in a high traffic corridor. If the population is that high, they are probably feeding on whatever they can find, which includes deer fawns.
Get out there and kill some! I would get as many as you can by shooting/sniping them first, and then go back in with the call a few days or a week later. Dont make any coyote call wise if you can avoid it, kill what you can first without calling!!
Every caller educates some coyotes, even if he never misses. We simply dont see every single coyote that comes in, and they can smell us from a long long ways off. Nobody hits every coyote he calls in, but that should be the pristine goal. Don't shoot unless you're dead sure of a good hit! It's always possible to come back another day and call that dog in with a different sound, but not nearly as possible or likely if you shot at him and missed after calling him in.
I would not be worried about messing up the deer hunt. If the dogs pack up, the deer will move out of there anyway, if they haven't started doing that already. It's getting wintery enough in most places by now that this years fawns can be smelling pretty good to the coyotes. Deeper the snow and colder it gets, the easier the fawns are to pick off.
Pull that trigger and get'ya some!
Good Luck