My married name is obviously different from my maiden name on my birth certificate. I always keep a copy of our marriage license with me just in case, however, I don't recall them looking at it, nor ever questioning it. Double check with Carnival, but as long as you have some sort of documentation I can't imagine it being an issue.
However (and I still need to do this) a passport would probably make all of that much easier. Also, ensure your birth certificate is an official State Birth Certificate - not the hospital one. If you are uncertain if that is what you have you can usually order fresh certificates from the state you were born in. If you search birth certificate and county and state of where you were born I am sure you can find the info needed.