If your cruise is coming up within the next couple of weeks, order a new birth certificate from the county. This will eliminate any problems at check in.
If your cruise is a few months away then I strongly agree with everyone else's advice. Go ahead and get passports for both you and your husband. With it you will have no question on what is required and what is a valid birth certificate. You will also be able to use it for 10 years of future travel.
The the identification requirements as of today are an original or state/county issued copy of your birth certificate and government issued picture id. That is "as of today". In today's world the requirements are changing on, almost, a monthly basis.