I don't know if you have any flexibility at all, but here's my two cents: I generally go to Disney World every other year, arriving on the Saturday after Thanksgiving and staying until the following Saturday. Which means that with Thanksgiving on Thursday, I'm there for the tail-end of Thanksgiving weekend.
I've never found the crowds to be overhwleming, as they are during Christmas week or the summer season. Saturday has usually been moderately crowded, but nothing too bad. Sunday you can tell people who've come down for the holiday are leaving... the various parks empty out as the day goes on, with people having to head to the airport and fly home. By Monday, the parks are much, much emptier. usually, by Monday or Tuesday I've been able to sail on to most rides without too much of a wait. heck, I've had Pirates of the Caribbean be a literal walk-on.
I guess what I'm wondering is if it might be possible for you to arrive on Thanksgiving weekend and stay into the following week.
I've also noticed that IOA and Universal are pretty empty this time of year. In fact, during my last visit, I started at Universal, hit everything on my agenda, then went over to IOA and did everything I wanted to, catching a bus back to the Wilderness Lodge and heading over to Epcot in time to catch Illuminations. was I exhausted? You betcha! Was it worth it? Without a doubt!
Richard