I'll jump back in here, since there's not much coming so far...lol! I've done a lot of reading on coyotes both here and other places. Since I don't have any first hand experience actually hunting coyotes I'll share a little what I've learned from others as well as my own experiences in the woods. You're right, coyotes are super cagey and smart. Coyotes that aren't used to human contact will bolt at the first hint of human, this comes from my own encounters with coyotes while hiking and hunting other game. So, my general sense with coyotes is you really need to not be seen, heard or winded. So, that usually means camo/concealment of some sort and some distance especially if the winds are not reliable. Again, my own experience hunting tells me that getting winded is likely to be the cause of missed opportunities probably 75% of the time and the other 20% is being heard and 5% is being seen. That's just my own guesstimates though. Coyotes have okay eyesight but are more adept at catching movement rather than having sharp, visual acuity. Their hearing and sense of smell are the ones you need to pay close attention to.
With all that in mind, if you go to Youtube and watch some coyote hunting vids, you'll notice most are hunting from a significant distance...most are certainly out of range for a shotgun. That said, depending on the wind and concealment I've seen guys call them in really close and thump 'em with 00 buck. So, that's why I invested a few bucks in an E-caller, the one I bought has a wagging flag/tail on top to catch a coyotes eye. With an E-caller and a remote you can position it where it's easily seen. You can then position yourself advantageously away from the caller giving yourself a good line of sight and hopefully a few directions you could shoot.
My sense is that an E-caller is likely my best chance at bringing one in...IF I can manage all the other factors too. It's kind of like baiting I suppose but check your local regs and see what's allowed. All that said, putting in the time to scout and look for sign is probably more important than a good set up. If you don't have any idea where they're at, the best set up won't do you any good.
Keep us posted how you progress. I don't know about you, but I come from a non-hunting family and I've had to learn everything from scratch, it's been fun but often frustrating. I am fortunate to have a buddy who wanted to learn to hunt just as much as I did and we've had each other to keep kicking the other guy's butt when discouragement hits.