“Why not load the cotton in New Orleans, ship it to New York, unload it, load it on another ship, and ship it to Europe? Wouldn’t that also be cheaper?”
I guess you never heard of Bermuda. Cotton — and rice, and sugar — loaded in, say, New Orleans was often shipped to Bermuda (part of the British Empire, don’t you know).
Can you point me to a source where I can learn more about these trade routes?