It's a case of coyote fever. We all get it. If you don't get it from time to time, then you might be someone that should hang it up for a while..
If you have the mosin, and a scope and can hit a beer can at 100yrds your fine. Especially in that area since the pelts on a good day will only bring $15 to $20, but that is a leap. Mostly they are only getting 10-15.
Skinning and stretching is a whole different step and before you get into that, I would suggest waiting until you can get a few.
Most of the shots are going to be under 200. Even for a new guy. In that area there are plenty of coyotes to cut your teeth on. If you want something other than the Mosin get the 223. Most shots are going to be under 200, and in some area's you are going to sit on the top of fingers out there and overlook a wash. Get some #4 buck for your 20 gauge and leave that on your lap. Get some shooting sticks and put the rifle off to the side.
Set out some kind of decoy. I prefer the Mojo, but a turkey feather will work.
for hand calls i suggest a longer Jack rabbit call, a cotton tail call, and a howler. Those three will keep you busy all season.
Since you are on a shoe string budget and already have a rifle and shotgun, then i would spend the money on those 3 calls, shooting sticks, and a seat. (with back rest) Avery Turkey seat works well and keeps you from sitting in cactus. Pads on the ground are fine, but at some point you will put it down upside down and learn why you don't use a pad.
As for area's.. from wickenburg to alamo lake and west is basically open to the state line. have at it. No point in burning gas to go to Williams if you can go right down the road and really learn the area around you.