350k or less for three bedrooms is going to be virtually impossible anywhere this side of the Anacostia, but if you can do it, it will be in Capitol Hill--solely because of that neighborhoods stock of smaller rowhomes.
I agree you are going to need to get closer to the stadium, or at least Potomac Ave metro to get something for 350k though. Also south of Pennsylvania and east of Eastern Market.
Have you looked at franklymls to see what is available in your range? FranklyMLS zipcode 20003 $0 to $350k