Don't go to the parks with kids on NYE! We went to Epcot this past NYE. We left the kids with the grandparents and got to Mission Space by 7:30 a.m. - the Fast Passes were already for after 2:30 p.m.!!! The were gone by 9:30 a.m. We went back in the evening around 6 p.m. and it was jammed beyond belief - not fun trying to push a stroller through the mass of people. I had made a dinner PS at the Rose & Crown to celebrate New Years in the UK at 7pm... there was a 10 minute line to check in, then another line just to get the pager... seated late, terrible service, cold food... I'll never do it again! After the early Illuminations show, we went back to our hotel (the Dolphin). A little before midnight we went down to the beach area of the Dolphin and watched the Epcot and MGM fireworks simultaneously - it was spectacular, hardly anyone around, and only a 2 minute elevator ride back up to the room. The next morning we were at the Magic Kingdom at opening (early entry if I remember right) and it was empty until mid-day.
My recommendation is book a room in the Epcot resort area, swim, ride bikes and/or play miniature golf on NYE day, and watch the fireworks from the Boardwalk or from your hotel.