American Airlines has round-trip flights from Dallas (DFW) to Washington, D.C. (DCA/IAD) for $179-$181, NONSTOP. Flights July to September. Mostly on Tuesdays, Wednesdays, or Saturdays. Origin/Destination reversible.
A Carry-on bag and also a personal item is allowed for free on American. You can avoid paying for seat selection by accepting the randomly assigned seat at the gate.