Turing NP complete but not Karp NP complete? Is there some examples of candidate problems that have Turing reduction from SAT but no known Karp reduction?

Conversely is there some examples of candidate problems that have Turing reduction to SAT but no known Karp reduction?
The complement of any NP-complete problem (including SAT itself) is polynomial-time Turing-interreducible with SAT but, there's a Karp reduction if, and only if, NP$\,=\,$co-NP.