We did January 1-4 back in 2024 and yes it was very busy, way moreso than we are used to in other months. That was to be expected but it was busy enough that we never plan to go back during Xmas or NY ever again. I don't have confidence that the crowds will go down as the days pass partly due to what Disneylover99 said but also due to the RunDisney event that starts on 1/7/2026. Back in 2024 there was also a race that started that same date I think, and that crowd was there the whole time we were too. They basically replaced the part of the crowd that left the first couple days of the month. Oh, and MCO was insane as you might imagine and we learned that a lot of the college bowl game fans fly in and out of there too which adds to the mess.
We used all the tricks to extend park time and get on rides without the really long waits so that part is definitely manageable. Disney Springs was awful though and gave a whole new meaning to wall-to-wall people. Men's bathroom lines were a half hour long if that gives you any clue.
So yes, expect big crowds but if you do your planning it will work. Pack your patience pills too!