You don't indicate where you are sailing from, but here is some information.
The last bridge before you get to the heart of DC that can be sailed under is the Wooddrow Wilson Bridge, which carries the Capital Beltway (I-95/495) across the Potomac between MD and VA. They've been working on replacing the old bridges with new ones for quite some time. Here is a web site that you can access, with information about clearances and numbers to call for more information.
WWB :: Keeping You Moving
The only large marina area I know of in the immediate DC area is in the Washington Channel, in Southwest DC. There are a number of marinas, including Washington Marina and James Creek Marina. There most likely are also areas where you can
anchor out. The Channel is very close to the Tidal Basin, Jefferson Memorial, and the Mall, either a short cab ride or a not-too-long walk. There are several restaurants and a fish market along Maine Avenue adjacent to the Channel.
Here is a web page for the marinas:
Marinas in Southwest DC
Wikipedia reports that James Creek Marina is 95 miles from the Chesapeake Bay. That's all Potomac River. At some point you may have to motor in the river, although it's very wide at the mouth and for some distance.