I spent a few days in the preserve some years ago and hit a lot of the dayhiking trails. The Sundew and Kirby Nature trails showcase a lot of the neat features in the park. I don't think there's a lot of backpacking that can be done as the preserve is a collection of long thin patches of property that go along the Neches River and Big Sandy Creek. Maybe kayaking would be a better option. I'd contact the park before you head out as I think some recent storm damage may have closed some of the trails.
Here are a few photos from the trip: Big Thicket National Preserve - TX - 051008 - 051108 - houstonkemper's Photos