Hi Brian,
Welcome to the Disney Parks Moms Panel!
Although I would not say that we have definitive "peak" seasons at restaurants, there are times of year when it can be harder to secure advanced dining reservations and/or the restaurants may seem crowded. Since you are dining on New Year's Day, the restaurants may be busy at that time. Because you do have advanced dining reservations, you should be fine! I would simply suggest that you arrive 10-15 minutes prior to your scheduled reservation time and follow the special instructions for checking-in at Disney's Spirit of Aloha Dinner Show.
Certain events can increase the crowds, however. Towards the end of your stay, the
Walt Disney World Marathon Weekend will gear up so you may see an increase in the crowd levels as runners start to arrive. Typically I can determine potential crowd levels by the difficulty I have making or modifying dining reservations. If I can still see some availability at restaurants, I am somewhat reassured. I would also suggest that you check the
park ticket pricing calendar for the dates of your visit. You can check to see which months are offering the lowest ticket price - for 2019 January is one of those months! This is good news if you are concerned about crowds. The new ticket pricing was intended to help manage attendance throughout the year to enhance the guest experience. For more information, check out this Disney Parks Blog article,
New Online Planning Tools, Date-Based Tickets Now Available at Walt Disney World Resort.
Brian, happy planning!
Magic, memories and Minnie Mouse hugs,
Donyell