Suggestions for state or national parks and kayaking

Good suggestons by taplep. Nantahala NF has several good campgrounds comparable to Elkmont,

You might consider Door County, WI (that's the "thumb" of WI on the map). Kayaking has been getting very popular there in the past few years.

I'm afraid the state campgrounds are probably already booked (Peninsula State Park and Potawatomi State Park) unless you are a tenter. If you are you might find openings. Peninsula is one of the most popular in Wisconsin. You could try a private CG, we love Egg Harbor Campground in Egg Harbor, WI (photo in sig is from Egg Harbor CG). Woodsy, but perhaps not quite like a state park. No water access from the CG though. However excellent hosts and most of the best of Door County peninsula is within 30 minutes of Egg Harbor. And its only minutes from Peninsula State Park and lots of other places to drop kayaks in the water.

Lots to do in Door County, biking, kayaking, hiking, dozens of restaurants (one is a decent brew pub), shops, lighthouses, islands, golfing etc. Almost no chain businesses north of Sturgeon Bay. Lots of state and county parks, there are 5 state parks, 2 with camping, 1 is a back pack only park (Newport), one is an island (Rock Island). Decent hiking in Peninsula State Park, Potawatomi, Newport, and Whitefish Dunes. Almost 300 miles of shoreline, with bluffs, underwater caves (Cave Point County Park), rocky shoreline (Whitefish Dunes) and a natural estuary (Mink River).

PM me and I can send you a link with photos of some it.

Our favorite is the North Bend COE campground on Kerr Reservoir, in Southern VA. It has a 50,000 acre lake, swimming, fishing, boat ramps, bike paths etc.. Check the recreation.gov website. Reservations can be made 6 months in advance, so you may be lucky.

Florida is surrounded by and inundated with water and good camping but I am afraid it may be more than 11 hours away.

While not a State Park Campground, take a look at Brunswick Family Campground on the Potomac River in MD(great kayaking, tubing and fishing). The C & O Canal is alongside the campground that has fantastic biking/hiking trails.
A short drive is a section of the Appalachian Trail and Harper's Ferry NP to enjoy History.A little further to Antietam NP. An hour from DC and Baltimore.
A couple of years ago management was taken over by River and Trail Outfitters and has improved from earlier years. Its not a resort campground but the fun and adventures are worthy. We live an hour away, hard to beat weekend or day trip experience for a variety of adventures.
From Indianapolis to Brunswick MD is 9 hours
potomacrivercampground.com

Drive 60-90 minutes over the GSM to Bryson City. Check out Deep Creek campground, Tsai bike trail (they have camping too), and Nantahala river for kayaking. That whole area is perfect for what you are wanting to do.
Tsali
Nantahala Outdoor Center