You might want to consider the Fish and Wildlife Areas. Some of them have campgrounds, and they are often less crowded this time of year than the state parks. (They are used a lot during hunting season.)
Pigeon River has a decent campground, and I have met people there who were canoeing, so I take it the canoeing is decent, too. Alcohol is permitted. You can't reserve a campsite in advance, but you will have no trouble getting one in the summer. You may even be the only one there!
In fact, you're making me want to go for a weekend campout myself.
