Birth certificates and driver's license for most married women will not match (for obvious reasons). The cruise lines are quite aware of that. However, since you will have your children's Birth Certificates with you, your maiden name will be on their BC's (problem solved).
Dave does bring up another good point though. If your ex has any custodial rights, you should have something in writing (notarized) that indicates he approves the trip. You may not need it, but you could be denied boarding if you get a particularly rigid agent checking you in.
As another safety measure - complete Carnival's "Fun Pass" embarkation online and you will be able to breeze through check-in without any hassles.