I'm not a camper any more, but I have friends who are and I have a Boy Scout. MN has some great state parks and maybe you can check them out online. Most of them require advance reservations. Itasca State Park is a little far north, but the beauty of that is you can see the source of the Mississippi River. Jay Cooke State Park just south of Duluth has wonderful hiking, but there is a river with a strong current so it can be a little dangerous for young children. You could look into campsite location. Savannah State Park near McGregor is beautiful. My parents have a cabin near there and although we've never camped there we've done a few day trips. There is a small lake you can hike around and it has a beach area appropriate for young kids. It has a picnic shelter with grills and a playground. It has a variety of campsites. McGregor is a small town about 10 miles away and has a few grocery stores, a DQ, a Subway, a pharmacy and other things you might need if you forgot something or ran into trouble. Aitkin (about 25 miles?) is a larger town that has a major hospital and a movie theater. Most people think of northern MN for camping, but I've heard there are also some beautiful state parks in southern MN and you don't need to drive as far. Savannah State Park that I mentioned is about a 3 hour drive from the TC. Of course, the North Shore just north of Duluth is beautiful too.
ETA: Savannah State Park is definitely in the woods and surrounded by nature. I only mention the modern conveniences of nearby McGregor and Aitken should you have a problem or need a back-up plan. I also just remembered that a family we know through school and sports used to do a lot of camping with a pop-up tent trailer. They said Savannah State Park was their favorite, although I know they also enjoyed Itasca.