Although I love Louisiana's relaxed atmosphere and great food, my husband had to sail without a certified birth cert. because they never sent the one he paid for and requested over a month before our trip. Mine took about two weeks to come from Dallas county. Also, we STILL have not received my husband's, and we've been back a few weeks.
Although they looked at all four certificates when I alone checked us in (my husband was returning the rental car), they did not bat an eye that his wasn't certified. It may have helped that the children's certs listed him as the father.
If you have the document attached that you described, I think you will be fine - have fun!