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According to the German Missions in the United States page on Schengen Visa / Short Stay Visa an appointment in Washington is not going to work:
Visas must be applied for at the German Embassy or Consulate General in charge of the applicants place of residence (for example an Indian citizen with H1B Visa residing in Portland/ Oregon must apply at the German Consulate in San Francisco; a Chinese citizen from Bejing visiting the US on a B1/B2 Visa must apply at the German Embassy in Bejing).
Please note that visa applications cannot be processed at a different Consulate or Embassy within the US or in the home country, if the person is residing in the US.