Looking to do a day of bouldering on my drive from Colorado to Chicago, around the weekend of May 15th.

The Dodge seems to have slightly better temps but a better chance of rain, so that's a factor. Looking for a good concentration of V5/6s, and possibly camping.

I can't decide! Both spots look good...

Patrick, the Dodge is closer to Chicago and is a really fun area, but overall the climbing in So Ill is amazing! You have way more options down there too: Holy Boulders, Jackson Falls, Drapers, Opie's Kitchen, Giant City State Park, etc, etc. For more check out Dr. Topo.com.
For just a days worth Gov. Dodge may be your best option and there is a good concentration of V5's and up spread throughout the park. Either way you should have a good time. Good luck.