Welcome to the boards. We have not been on the Triumph, so I don't have an answer for your question. But I am interested in others answers since we will be on the same cruise with you along with my in-laws. We are also almost neighbors, just outside of Memphis.
I have cruised out of New Orleans before. The trip down the Mississippi to the gulf is a great trip in itself.
Of the three ports, I've only been to Nassau. Didn't go to Atlantis the last time and have regreted it. So that's our plan for that port.