Collier Seminole State Park can do all that... it is just a few miles to the west.
And add on 15 miles of wilderness to the Wilderness Waterway, make your first stop Tiger or Picnic Key.
I always bring my own shower: http://www.coleman.com/beverage-jugs/5-gallon-water-carrier/5620B71...
As for parking. that's up to you. Another alternate for the north part of the 10K is launching at Goodland.