It's been fairly full the two times we've been there. But regardless you are talking about 2 weeks from now, so that explains it more.
There are only 2 TS restaurants in the AK that accept the Dining Plan. I also was surprised at first when I tried to get it for our upcoming trip, and was only able to get 11:00AM for lunch. But there is only Yak &Yeti and Tusker House at the AK that accept the dining plan, and esp. because we are going during FD, it books up quickly.
However if you really want to eat there walking in is a distinct possibility.