You should be able to keep the Blue Cash open and open the new card. Then have AmEx backdate the new card. I've heard it's possible.
Current Cards:
American Express Blue Cash Preferred - March 2007 - CL $10,000
Citi Forward Visa Signature - September 2009 - CL $6,500
Chase Freedom - March 2011 - CL $15,000
Chase Sapphire Preferred - May 2013 - CL $12,500
Discover It - November 2013 - $5,000
Barclay Arrival - January 2014 - $10,000