Defintely do the new booking before canceling. I had an Embassy Suites episode similar to several FTers above before getting wiser.

Another lesson I learned, sometimes it is cheaper to book several one day reservations at the same hotel. The rates sometimes are cheaper, on the weekend for example. If you ask for a 5 consecutive days on one reservation, sometimes the quote is 5 days at the same (higher) rate. Other times the rate detail window does show rate change, so check it out to see if it make any difference.