You can call DME directly at 866-599-0951 and tell them you want to book DME from airport to resort at whatever time you’d like to go. Have your resort reservation information at hand, since they need that to book your transfer. That way you won’t need any flight arrival information.
Do not call the Disney reservation line or use the web reservation form, since these require flight information and many of the Disney reservation agents only know how to complete the reservation form with that information being completed.
If you’ve already booked DME for the morning of your original planned arrival, you don’t need to do anything, except you should not use the yellow DME bag tags since you’ll want to retrieve your bags and take them to the Hyatt when you arrive. I have read that you can have Hyatt bell services retrieve your bags for a fee, but do not know any details about this.