You might want to check the Sheraton World Resort. I don't know if it's within 15 minutes of the Coronado Resort, but they do have complimentary transportation to WDW.
Have you tried calling the Platinum Concierge Desk to see if they can get you a room.
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Platinum Preferred Guest Benefits
2. Platinum Preferred Guests will be guaranteed one room for personal use only for reservations made at least 72 hours prior to the date of arrival. Member level upgrade is limited to one room per member staying at COR rates or SET rates with last room availability contract. Guaranteed availability will not apply at resorts or other select properties or during publicly recognized events.
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Edited to add:
I noticed the following on each of the WDW Swan and Dolphin pages, so you might not be able to use the method stated above, but I guess it's worth a try.
<font face="Verdana, Arial, Helvetica, sans-serif" size="2">As of December 29, 2002, there is a Resort Service Fee of $10.00 per room per night. The fee includes health club membership, USA Today newspaper (Monday - Friday) and a high-speed internet connection.
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[This message has been edited by kluau88 (edited 02-25-2003).]