The Traverse City State Park is almost in downtown Traverse City ! Not the place to go for QUIET, but they do have a very nice beach ! August is still "prime" vacation time, but the last week things really start to slow down. Many places are DEAD after Labor Day.
A bit farther north on the west cost is Wilderness State Parks. The have several camping areas, including some that are "rustic" (pit toilets, no electricity).
I have not been there in a quite a few years, but along US2, just west of the bridge, there are many pull out right next to the beach. They used to allow overnight dry camping. Someone needs to update this ! Everything else you requested is there.
There are DOZENS of state and national forest campgrounds. Almost all are rustic.