I've noticed that the hotel website can go a little wacky sometimes. this has also happened to me before, it's just a glitch in the system. I would give it a day and check it again.
I had trouble getting the room I wanted last December and everyone kept saying to wait and the rooms would show up. Turns out there was a big convention in town and the rooms never opened back up.
Try changing your dates around to see if that makes a difference. When supply gets low, I think they hold back rooms to use for longer stays. I was not able to book 12/6 arriving on 12/5 or 12/6. But eventually someone pointed out they showed it available if they used an 11/30 arrival date. I was able to book with that date and later drop off the dates I really did not need.
That time gets booked quickly! You can keep calling Disney to check for cancellations but chances are it will stay full. It is extremely busy at that time of the year. I have stayed at DLH twice (in the past) during this time of year and everything is full and crowded! But if you have flexibility trying staying right after. I have stayed after Labor Day and it was so nice!
i prefer to call when i book anyway. That way you talk to someone in real time and you can put down what you are celebrating or make special requests and they note it in the reservation. Plus you never know what magic may happen when dealing with an actual CM