I won't comment on the weather because it has been well (and correctly) covered. As far as finding open campgrounds, I know of 2 that would be open. Timberlane in Clarkboro NJ is very close to Philly open year round and Capitol KOA in Millersville MD is convenient to DC, but doesn't open until March 1st.
As for VA, the Fredriscksburg KOA is open year round. I really like this place; we make it a point to stop here for a couple of nights on the way to and from. Close to Richmond and various Civil War sites.