Funny to find you posting on my local site when I just recently posted on yours..
As noted by the Leprechauns, there are a number of virtual caches downtown by the monuments. Try searching for caches in the 20005 zip code.
If you want to try actual caches consider "Neglect" and "Rose Park" in the Georgetown area.
If you have a car and can get into Northwest DC, "S Valley" (My favorite cache so far. A small wooded area with an established path in area of embassy residences and other stunning houses.), "DC Line" (This is were I left the TB heading for the UK), "Sibley" and "B&O Railroad" (This is cache is on the "Crescent Trail" a mostly paved path that runs from Georgetown, in Washington DC into Maryland) are all within a very close driving distance from each other. You can map them by doing a search for caches in the 20016 zip code.
If you have someone at home who would like to see you live from DC, consider "*Washington, D.C. Live."
Sorry, but I don't have any recommendations for Virginia - - still new to the game.
Safe travels.