You might want to try some of the Indiana State Parks.
Turkey Run (west), Brown County (southeast), and McCormick's Creek (southwest) would all fall within your mileage range. As of a few minutes before I posted this, Brown County and McCormick's Creek still had sites available for Labor Day. Turkey Run, which I'd recommend because they have canoe trips in and around the park, had ONE site available when I checked. All three of these parks are nice. There are probably a couple more within your mileage range as well but I haven't had personal experience with any of them.
Tom (Who is leaving for 4th of July weekend at another Indiana State Park [Potato Creek] in about an hour)