I don't know how it works if you're dealing with Disney directly, but I'm a TA and I have this happen all the time. I just call and ask for a couple of extra days. They are very accommodating, but you'll want to make sure you let them know.
Before Disney cancels your reservation, you'll hear from them directly at whatever phone number you gave them when you made the reservation.
You can PM me if you want more details, but that's basically how it works.
Have a good trip.