Not to throw a monkey wrench into things, but if you booked a package with the dining plan, can you upgrade to AP? I was under the impression that in order to have the dining plan, you had to book a package that includes at least one day of MYW tickets. If you upgrade, you won't have MYW tickets anymore, therefore, not eligible for the package. Unless you left one day of MYW tickets on the package. Does that make any sense?
Anyway, once you have your AP number in hand, you can call DDE and purchace the DDE card. You can tell them you are already at WDW and they will have a temporary card for you at Epcot GS. However, it may not be there that day. I have heard that it sometimes takes 24 hours for them to process it. So, make sure you don't have any OOP TS's during your first 2 days of your trip. Hope it works out for you!