I presume that you would dismiss out of hand the following answer to your first question: "Yes, there is a single genuinely correct, all-purpose logic, and there is no such logic, and there is more than one such logic." So I take it that your question presupposes that no correct logic could allow that answer to be true.
If you're asking whether there's any good reason to abandon the standard, two-valued, "classical" logic routinely taught to university students in favor of some non-classical logic, then I'd answer no. Some philosophers say that we ought to adopt a non-classical logic in response to such things as the Liar paradox or the Sorites paradox, but their arguments for that conclusion have never struck me as persuasive. I think that the Liar and the Sorites can be solved using only classical logic (and bivalent semantics), or at least it's too early to conclude that they can't be.
For a much more detailed answer, you might consult Susan Haack's book Deviant Logic, Fuzzy Logic: Beyond the Formalism (University of Chicago Press, 1996).