The crowds for New Year's Eve are very heavy and everyone is partying hard that night. It's a great atmosphere! I would strongly suggest getting to CityWalk early in the evening to try and beat some of the crowds. January 1st during the day isn't too bad in the parks but still busy.
As Metro said, it's pretty busy at night on the 31st. If you want to stay on site, keep trying. Rooms open up often and it will definitely be a plus for the parks.