What does your lease agreement say about subletting? There is no inherent right to sublet--the lease itself controls. Most lease agreements give LL right of refusal--which your LL has now apparently exercised. Your situation will turn on the plain language of your lease. Bear in mind if LL refuses to allow another roommate you MAY be required to come up with the full amount of rent or face eviction.
If you want chapter and verse on the law you'll have to hire an attorney. Most (if not all) of it can be found online--the DC bar has links for landlord tenant laws in DC.
READ THIS BEFORE CALLING OR EMAILING ME: I am licensed to practice before the state and federal courts in Virginia. We have not established an attorney-client relationship unless we have a signed representation agreement and you have paid me. I am providing educational instruction only--not legal advice. You should speak with an attorney to whom you have provided all the facts, before you take steps that may impact your legal rights. I am not obligated to answer subsequent emails or phone calls unless you have hired me. I wish you the best of luck with your situation.