As stated, late August is a good time, but be sure to avoid the Mackinac Bridge area around Labor Day weekend. There are a LOT of tourists in that area for the annual Labor Day bridge walk.
My preference would be northwest lower Michigan and areas along Lake Michigan. In fact, we're heading up that way in a couple of days for about 10 days. We'll be camping at Orchard Beach State Park in Manistee and then heading up to the Sleeping Bear Dunes area. The Leelanau Peninsula (the little finger area just west and NW of Traverse City is my favorite area).
There are some nice state parks along the Lake Michigan shoreline from South Haven (Van Buren State Park) to Manistee. They do get busy, especially on weekends throughout the summer.
We don't have a lot of experience with private campgrounds along the lakeshore, but click on RV Park Reviews.com below and you can search for some that might work for you.
http://www.rvparkreviews.com/regions/Michigan/