Hi - I work in animal welfare, so that's informing what I say.
There's no magic bullet here. You can find someone to take the cat (a friend, family member, neighbor, etc).
You can reach out to local rescue groups and see if they can take your cat in - King Street Cats is a good one; there are others in the area, too. But these are groups with limited resources and can't take every animal.
Or you can bring the cat to a shelter, where he or she may or may not make it out again. DC area shelters are by and large really good, so your cat has a better shot there than in other parts of the country, luckily. But cats still have the highest euthanasia rates of all animals, and this is a very tough time of year for cats in shelters - we're still in kitten season - so, there's that.
There's a new website that's supposed to help with rehoming options. I don't know how much use it'll be since it is so new, but it might be worth a try.
http://pawslikeme.com/
And those are your options, basically. If you go the rescue or shelter route, please offer to make a generous donation to help cover the expenses of caring for your cat.