You cannot just show up with a baby and a credit card. Both parents must appear in person. Think of it as a pop quiz for your relationship. If one parent can’t come, you need a notarized form (DS-3053) and a photocopy of their ID. It’s easier to just drag them there, honestly.
Bring the original birth certificate. Not a copy. Not a 20-year-old hospital bracelet. The certified, raised-seal, official one. You will also need a list of travel dates for the next year, even if you’re just going to grandma’s house. They make you guess. “Um, maybe Mexico? Who knows?” It’s fine. Be vague.