I believe Manistee National Forest is dog friendly. I know the fire trails off the main road are dog friendly but I'm not sure about the state park campground. The beach by the campground is dog friendly though. We took ours there last year.
We stayed at muskegpn state park and drove to a dog beach call norman f kruse I think it is called. No state parks have a dog beach in michigan but this park but you cannot camp there. The dog beach is really nice well kept they even supply doggie bags.