I don't know if it's in your budget, but we did Wild Africa Trek about two days after Christmas a few years ago and it was one of the best decisions we made for that week. While AK was packed, we were on the safari, during the tour, with a smaller group having a great, immersive experience. Other than walking through the park after the tour (we did one fairly early in the morning), we never would have known the park was packed to the gills. They don't hold many tours during that week (due to crowds, I assume), so, if a tour interests you, you may want to check the schedule to see what is available that week.
Unfortunately, even the less popular attractions are crowded during that week and getting around the parks can be difficult. My only other advice would be to avoid, particularly, the mid-section of EP WS during Candlelight Processional times. We found CP show times were the best times to tour Future World for less crowds.
If things get really bad and you want to leave the parks, you could always do a monorail tour, tour the Boardwalk resorts (we like to do this at night so we can see the entertainment on the Boardwalk as well) or take a boat to Ft. W from MK. Not in a park and it did cost extra, but we spent an entire afternoon at Ft. W on a Christmas trip - we went to the first HDDR and then did a private sleigh ride. Super enjoyable and no crowds!