OK. Post one legitimate historical reference to the taxation of cotton exports from the US and I will admit you are right.
AFAIK, exports from the US have NEVER been taxed. Tariffs are placed on imports.
The southern argument of the time was that since (untaxed) southern exports of cotton were what primarily paid for the (taxed) imports of tariffed items, the South was “really” paying for the imports.
I’m not an economist, so I’m unclear to what extent this argument is legitimate. I do know that to the extent it’s legitimate, it’s about the balance of trade between nations, not about whether an individual exporter is paying taxes.
Prior to the War, northern business interests were concerned about the effect of stopping cotton exports on the balance of trade. As it turned out, they didn’t have to worry. Union exports of manufactured items and agricultural produce increased dramatically during the War. (And none of those exports were taxed.)
I have posted several, including a NY Times article (pdf file) dated April 1861.