You should be able to get a plenty nice room in DC for well under $200, almost anything but the top-end luxury places. March should be pretty dead. I've been there in March, and I bid on Priceline and had 3-star Renaissance Convention Center for 50 bucks/night. This was March 2002, granted. Anyway. Some zones have iffy 3-stars (in my opinion) in DC, but you should be pretty safe in any zone (choose your favorite) with 4-star bidding. Some of them are kinda bland. My last 4-star win there was the JW Marriott, and I don't particularly care for that one, though for sightseeing it's a great location, and it was under $100/night.

Bidding 4-star for Foggy Bottom should get you Park Hyatt, Westin Grand or Melrose. (The Watergate has come up but not since ending association with Swissotel.) Downtown-White House should net you pretty nice 4-stars as well. I mentioned that I was pretty unimpressed with the JW, though, which seems to come up a lot. You can get the Willard that way if you're lucky, but I wouldn't bank on it.

Couple other things to try if you don't bid or want some comparison pricing:

QuikBook has a nice selection for DC, mostly independents rather than chains. Prices are sometimes quite good, other times just so-so, but again I think March should be pretty quiet so may have some good rates. Check their prices for the Willard, Madison, even Hay-Adams, you never know what you might find. http://www.quikbook.com/10031.asp shows you all their hotels in DC.

Capitol Reservations is a local discount reservation service for DC. Can have some very good rates on properties that they do list, but the rest of the list could be so-so or even above average. Worth checking, though, on the chance they have a good deal that works for you. Has some of the same hotels as QuikBook, so check again for a deal at the Hay-Adams, etc. http://www.capitolreservations.com