You are doing it right, I have to say. You've done a lot of research and tapped into a good resource here.
I can't help much, but I'll say what I can:
I am in Oklahoma, not Illinois, so I can't tell you any specific locations. But I'll bet dollars to doughnuts you've got farmers or ranchers in Illinois who would be tickled pink to let you have the chance to whack 'yotes on their land. Coyotes do NOT help livestock. So ask the guys with the livestock.
I have no idea what a habitat stamp is.
You, and our Brethren here on this board are pretty amazing sportsmen. I commend you all. I've never done anything more with a freakin' ugly, useless, livestock-stealing, game-murdering 'yote than urinate in the exit wound, curse at it, and leave it lay.
You're probably going to have to get license for each State.
I would if you were here or if I was there! I've killed more coyotes while deer and turkey hunting than just hunting the 'yotes themselves - - buy you can sure hunt them themselves. And it's fun
You are on the right track! Learn, and HAVE FUN