There are a few options. First, you could leave your car at National Airport. It's a $10 max if you park in the economy lot. You could take the blue line into town and back out to your car.
Also, you could take the GW Parkway into Rosslyn (Arlington, VA, just across the river from Georgetown) and see if you can park in the Hyatt's parking garage for the day. That's right across the street from the Rosslyn (blue and orange lines). There should be a bunch of parking garages around there, and you wouldn't have to go into the district. Also, the blue/orange line runs right along the mall, so no changing trains.
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Also, you could take the GW Parkway into Rosslyn (Arlington, VA, just across the river from Georgetown) and see if you can park in the Hyatt's parking garage for the day. That's right across the street from the Rosslyn (blue and orange lines). There should be a bunch of parking garages around there, and you wouldn't have to go into the district. Also, the blue/orange line runs right along the mall, so no changing trains.