RE: Any suggestions
IL has some good public hunting in the Shawnee National Forest, located in the southern 1/3 of the state. You could get an application in by early Feb. to be included in the 3rd lottery. IL season is split into five seasons and that time frame would be the middle of the 3rd season and the start of the 4th. You would be very lucky to draw the 3rd season and would probably be better to draw the 4th season to try and hunt on the way home from Michigan. Tags are given out by county so you would need to look at a map to determine where to apply to be in the National Forest. IL DNR site will be showing the tags that are left over after the 1st and 2nd lotteries by sometime this week. You can print off the applications from the web site. Il tags and lic. have went up this year and is probably a bit pricy, but it's something to consider. I think you can apply as a group so that if one gets drawn you all get drawn. I have killed birds in about 4 different counties in the National Forrest and they have come from any one of the seasons. The split seasons help it to not be quite as crowded, and the later seasons are normally not bad at all during the week. It only takes one gobbling turkey to make a hunt if your lucky enough to find the right bird. If you decided to try for a permit just email me and I could try to point you in the right direction on finding maps and what I might know about the counties that have tags available.
Missouri also has a ton of public ground and a lot of turkeys in the southern portion of their state. I believe that is about the time their season is in. The state will send you maps of the public ground and harvest records if you want to see them. Unless it has changed I think MO tags are over the counter and not near as high as what IL has went to. I don't really know where to tell you to hunt, but the harvest records along with the maps can point you in the right direction.
Good Luck