I live in Adams Morgan and just went through trying to find housing. It's a bitch.
Adams Morgan is great, but you'd probably need to find roommates because so much of it is row houses. Which is a lot of DC, actually. But I would try H St NE (it gets less grimey headed south toward stanton square), Petworth, Ledroit park, Along first st NW (I forget What that neighborhood is called, Truxton circle?), and maybe Columbia heights but that's getting expensive too.