Every northern Illinois public land that I am aware of requires you to draw gun permits there via the lottery. 95% go in the 1st lottery. It's a real crapshoot for them. I moved to Illinois 5 years ago and drew 3 of those years, but had to be lucky I think. The only public land I know does not require drawing is the Shawnee NF down south. For there you just apply for that county's permit, which generally are available even after the 1st lottery.
If you must stay north, I suggest applying for a less popular property, but even those are hard to identify, because the ILDNR does not publish any draw stats for any public properties.