Hi Athenna...
Is your 'closing date' with Disney or on a resale?
If it's a resale, you sign and pay and then the seller signs and accepts payment and then the escrow/title company sends the deed off to be transferred into your name. They also inform Disney and then Disney says to give 2 weeks to get into their system, but they are sometimes faster.
I did the whole process by email, fax and wiring...fairly painless and 5 minutes of my time.
Don't know what Disney does, but the 'closing date' is really the day that the money changes hands from you to DIsney.
Congrats!