The Swan and Dolphin are not Disney properties (I'm sure you knew that) and rely almost exclusively on convention business. Having planned one I can tell you it is not unusual to commit to convention dates two or more years in advance. So it is possible that the hotels are full for that week.
Also, other countries don't have the same school breaks and holidays that we do so there may be a foreign group already holding rooms for that period. We stayed at the Dolphin Christmas 2001. We had booked in June and got the government rate. When we checked in, the bellman told us that we would see a lot of Japanese tourist during our stay because a tour group had checked into more than 1,000 rooms at the Dolphin and several hundred at the Swan a few days earlier, but would be leaving the following Sunday.
We thought the AP rate was still too high for us over President's week this year so I tried for two weeks to book the Swan or Dolphin and couldn't get more than one night of our stay at the government rate at either hotel. I was about ready to give up and pay the higher AP rate one night while posting to the Dis and decided to try DU and see if they could get the government rate. Afterall, there was no charge if they couldn't. I received an e-mail that they had it for us but I was sweating bullets 'til I received the official Swan's confirmation number via e-mail a couple of days later. When we were checking in, they did ask for proof of government employment so I showed them my State I.D. badge. I also overheard one of the desk clerks tell one woman that they were 100% booked and couldn't give her a room for even the night and recommended she try some of the hotels on Hwy-192.
So if you can't get it with the S/D reservations, give DU a try. They might be able to make the reservation for you despite what the reservationist are telling you. There is no fee to use them and you might get lucky.