Consider one of the several state recreation areas between US-23 and US-127, north of I-94. Brighton, Island Lake, Pinckney, Waterloo. That part of the state is rich in camping facilities in public and private campgrounds and RV parks.
Check the Michigan State Parks web site for what kinds of sites are in each of the several campgrounds for each recreation area. Go to the campground map and pick Jackson County to get into the middle of the area.
If the campground has electricity for a RV, chances are it may not be as rustic as the primitive camping areas, but might still be OK for tenting. You'll have to do some of the homework yourself, only you understand just what it is that you want.
There are also many RV parks in the Irish Hills region, but facilities will be more like a RV parking lot, not so camping friendly as the recreation areas.
Tom Test
Itasca Spirit 29B