Your old card isn't already deactivated? That usually is done automatically when the new card is issued.
Definitely use the new card. It shouldn't be a non-issue. Your account wasn't "cancelled" -- you were just issued a new number for it. Your flights are already paid for right? They (and anything else that is prepaid) will be no problem at all since the charge is already processed. Nor will anything that would be paid for at check-in -- just give the check-in staff the new card at check-in and have them run the charges on that card.
The grey area is anything that is scheduled to be automatically put on your credit card in the future but not at check-in. For those things, just call the companies/log in to your reservation and update the billing info. But kind of the worst thing that would happen is that the company tries to put the charge through, it doesn't go through, and you get an "excuse me, but it appears that your billing info is no longer valid" email and have to call/log in to update.