Can you be more concrete about this? How would you resolve this specific paradox without throwing out any obviously true axioms or introducing any obviously false ones? If it was easily resolved, it wouldn't be called a paradox.

Banach-Tarski is a paradox. You can resolve it by deleting the axiom of choice. But the axiom of choice is obviously true, at least as much as B-T is obviously false. That's why it's a "paradox" and not just "a proof that the axiom of choice is false"