I'm not really much of a camper but I know there are a bunch of nice sites set aside for camping in the DH Day National Lakeshore area around Sleeping Bear. Your tax dollars bought that valuable land so you might as well use it.
I think you ought to consider going to North Manitou island to camp. It's neat in that it very rustic. Nobody lives there. There are no varmits like skunks, porcupines or racoons. It's a big island with beautiful beaches and sand dunes.
Start your research here:
That's who will take you there. Once they drop you off, you're stuck there until the next boat comes. I don't know if they go every day or every other day.
Bring sweatshirts. Depending on your body temp tolerance, you may or may not be able to swim. I only brave Lake Mich from July 4-mid August. Water temps usually max out at about 72 degrees
My wife spends all summer up there and I go back and forth when I can. In fact this is where I get to use my "Sea Ray". Wonderful place to escape in the summer. I've never understood why people go to Florida in the summer...