I know I'm digging up an old thread. I was looking for something else, ran across this and noticed taht the question wasn't answered.
Try Battle Row Campground in Old Bethpage. I stayed there for about 10 months while I was working on the island. They aren't full hookup - only water and electric, but have a place you can dump on site but have to move your rig. They turn off the water to the sites in the winter, but keep water flowing at the site where you can dump your tanks so you can fill up there.
It is right on the edge of Nassau county at the Suffolk county line. Suffolk county has rules on length of stay (7 days I think and then you have to leave) Nassau has no such rules and allow you to stay as long as you like.
Here is a link to their website:
Nassau County Department of Parks, Recreation and Museums