I went through the same thing when I first moved to the area. When I eventually found a primary care physician, three years after being here, he was terrible and all the way out by Landmark Mall. I don’t know why the shortage of doctors hasn’t been brought up as a public health concern; unless you’ve lived in DC for over a decade it’s nearly impossible to find one that will take you.
Last year I was thrilled to come across a site that will identifiy doctors taking new patients that accept your insurance. It’s great! http://www.zocdoc.com/
Through ZocDoc I found Dr. Natallia Shotashvili, who is located right across the street from the Woodley Park metro. I’ve only seen her once but was happy with the visit. Although she seemed a little rushed she had a good bedside manner. She takes United Healthcare and her office hours are really convenient.