Hi! My dh just gave me a last second trip for my birthday (was Tuesday). It was very difficult finding rooms. We ended up getting rooms on travelocity splitting our six night stay between AS sports and PORR. I would call Disney first because the online sites have to turn back in any rooms not sold. The other option is to go online to the swan or dolphin web site and book directly from them (especially if you are a teacher or gov't employee, can get $149 / night!) You can also go to priceline and put in a request for a resort in the disney area with a good bid (around $80 - 90 should do it). You most certainly will get a wonderful resort, and probably the Dolphin. We have stayed there and loved it.
I found that travelocity was the best online booking engine for a non disney site. They don't charge big fees and there is a $25 change or cancelation fee.
Our dates aren't until July 9. We are going to see the shuttle launch. We have aps so it really is still a budget trip for us.