Hi crispy, and welcome to cruise-chat.
It's true that Celebrity is more formal than Royal Caribbean, yet not overly so. While you might find some gentlemen in tuxedos on a 10-nighter from NYC on formal night, it's not a requirement. Note that you'll have to be at least "smart casual and above" in the show lounges every evening -- but that's not a big deal, IMO.
Enjoy your first Celebrity cruise. If you're like me, it won't be your last.