The North bought it's cotton from the south, and I'm unaware of any source that would indicate that the Northern industries were starved for supply.
They raised tariffs on the cotton so much that the South could not sell it easily. That was one of the main reasons for the animosity that the South felt toward the North.
I would direct you to Article I, Section 9, Clause 5 oc the Constitution which reads, "No Tax or Duty shall be laid on Articles exported from any State." How could there be a tariff on cotton exports when the Constitution clearly forbids it? Do you have anything which supports this claim and which might state what the duty was on these cotton exports?
When did any thing being unconstitutional ever stop the Government from doing as it pleases?