You’re exactly right, Kerry — it’s a very “eat local” food that spoils very quickly after ripening. That’s why you rarely see them in grocery stores. We used to have so many that mom and grandma would have to make pies, cookies, etc. to not waste (they were very militant about not wasting anything).
I’m honestly not sure if they would grow in your area, but I did find this fact sheet from Perdue (the University) that may be of help. I’ve also attached a growing map above I found, via the USGS Geosciences and Environmental Change Science Center, that indicates the growing area. Parts of Illinois is included on the map, although I am not sure if your particular area is.
A few other notes: You can buy paw paw seeds on Ebay, although, in my opinion, $12 for eight seeds is ridiculous. (Prices may have come down, I’m not sure; this was just pointed out to me by a friend a few months back.) Another thing: I have heard (though never experienced first hand) that the taste of the paw paw grown in the northern areas tastes a bit different than down south. I’m really not sure why. If you’re going to grow (and I really hope you will, because it’s an amazing fruit!) then just be sure to remember that the fruit spoils and ferments very quickly after ripening. Oh, and it makes some FANTASTIC beer. My uncles used to make it.