What is your preferred mode of transportation? Since Georgetown doesn't have a metro stop, many commuters choose to use the bus system, if they don't have access or desire to use a car. Also, Georgetown is one of the most expensive neighborhoods in the city, so the further out you are, the more space you can get for your budget.
I'd start by looking into the Tenleytown, Shaw, Cleveland Park, Rosslyn, and Petworth neighborhoods to start.
You can contact me at (202) 505-4105 or at Molly@citylightsrealtygroup.com. I'd be happy to send you listings within your budget.