Definitely keep craigslist open for last-minute grabs. Be prepared to call people and do walkthroughs day-of postings. Some other possible sites are http://www.apartmentguide.com/apartments/Washington/Seattle/ and http://seattle.apartments.com/.

Another way to stay ahead of the game is by contacting realty companies for larger complexes that interest you, and ask them about move-out dates or preleasing.

I don't live in the Seattle area, but for all my moves, I've used rent.com and apartmentratings.com. It really depends on what you're looking for, but if complex living is okay with you, both those places have good leads and generally a lot more information than you'll find on, say, Craigslist. I like Apartment Ratings because you can find personal experiences from people who have lived in different places. Just as an aside, rent.com offers a 100 dollar reward card if you find a place through them. I found the place I'm in now on their site and did get the card, so that was a nice little moving-in bonus.