Then why did exports flow from southern ports to Europe instead of being sent from there to New York and from New York overseas?
I believe that is mostly what happened. Unless you choose to believe ships traveled from NYC empty to load cotton in southern ports. Or that they traveled empty from Europe to southern ports.
I believe most cotton was carried by coastal ships from the South to NYC, where it was loaded onto ships to cross the Atlantic. Presumably those ships loaded a good chunk of the imported goods to travel back to their southern ports.
NYC was the #1 cotton export port, followed by New Orleans and Mobile, but I think at quite some distance.