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PeaRidge: "The merchants of that city imported three fourths of the European goods consumed in the country, and they in turn exported nearly all of the great crops with which the balance of trade was maintained."

Once again, this report is from New Orleans, 1857:

"Of cotton the total receipts at New Orleans for the year were one and a half million of bales, the average price being 124 cents per pound.
Of this. there were

shipped to Great Britain 749,485 bales;
to France. 258,103;
to other parts of Europe, 186,069; and,
to United States ports, 223,204 [=15%].

The entire crop of the South is estimated to have been a little short of three millions of bales."

So New Orleans claims to ship half of all cotton, 85% of it directly to Europe, and again, this map demonstrates that upwards of 80% of all cotton must ship through Gulf Coast ports:

Therefore, it appears that claims all cotton exports went through New York are highly exaggerated.

Imports are a different matter:

Here we see in 1859 that 75% of import revenues did come through New York, and over 90% through the four largest eastern cities.

The obvious conclusion is that ships exporting cotton directly from Gulf Coast ports to Europe returned with imports to New York, where they paid Federal duties on those imports.
But in order to ship millions of bales overseas, those ships numbered many hundreds, meaning dozens in any given Southern port, some owned by New Yorkers, others owned elsewhere (but none foreign).
So cotton producers had choices on which shipper to patronize with their business.
Nothing prevented them from choosing Southern shippers.