As a hunter, I always hate these threads because invariably somebody will recommend any number of things that will kill indiscrimately, inhumanely and illegally.
The best way to get rid of coyotes is to (legally) trap them. They are readily trapped if you know what you are doing and can be trapped year round. It is INFINITELY more effective than shooting them. If you are trying to shoot them in your backyard, you will have done VERY well to be able to shoot more than one. They are canines and they are SMART. They are not likely to be fooled twice, I know, I have the same "issue".
A remote control electronic call, of any kind (provided it has even decent sount quality) is better than having the dogs come in looking directly at you. I have an older Foxpro but have had success with even a cheap handheld ($20) call.
They WILL circle downwind, EVERY time, book it. Set the call up as far upwind as you can but with a clear shooting site to within 50 yds downwind of the call and have it. Good luck.
No coyotes are NOT native to Georgia and while they may have interbred with the once native Red Wolf as DNA sampling suggests they are in fact a different critter WAY more prevalent. As little as 25 years ago they were not only not in every county (as they are now) but seeing one at all in most of Georgia was something to talk about. They are absolutely everywhere now including all of rural, suburban and even urban Georgia.
I also (legally) shoot every blasted one I see.
Just curious... what is an illegal way to shoot or trap a yote? I had a long discussion with my local DNR officer and he was very clear when he advised that you can kill non-native species any way that you see fit and as long as it doesn't affect any other native animals. He went on to say that not only are you allowed but it is encouraged to get rid of these Fawn and Turkey Killers.