I just returned from a motorcycle ride over to Pohick Bay Park which is located in Lorton VA (northern VA). They are 100% hammock friendly and in fact told me to figure out which trees would work best with my hammock and they would let me have that site. Only restriction was that tree straps are required including for tarp hanging.
Rates are not cheap. 27.+ per night for non electric site. This is not primitive camping and they do not have primitive camping. They have a lot of things to do. Here is link to the site. http://www.nvrpa.org/