The week after NY the parks are really quiet from the first Monday eg. this year, Monday is the 3rd of January so we anticipate quiet in the parks here on in.
It can be very cold or warm; in other words totally unpredictable weatherwise although there will be less rain. 2 years ago we had 2 short showers in the whole first 2 weeks of January we were there. You should still be able to catch some of the Christmas decorations in the resort hotels and I know that these are normally at cheapest price this time of year.
PS in restaurants is not normally necessary and you can pretty much walk in anywhere, with a few exceptions of course (CRT). I think it is a great time to go as you can get about without big crowds in your way and get on all the rides you want with hardly any wait.
I will say that park hours are shorter, more rehabs are likely to be taking place and it can be downright chilly in the evening. Other than that, absolutely go for it! We'll be there jan 1st to 14th 2005 and can't wait.
Maybe my trip report when I get back will give you an idea (or other trip reports for this time of year)
Much pixie dust,
Claire