Natural RE undecidable problems but not Turing-complete
There indeed are sets that are undecidable but not Turing complete.  However,  I'm not sure there is such a thing as a natural undecidable not Turing complete language.

Emil Post set out on a search for exactly this (See Post's problem) and his research resulted in the Simple sets.  To quote from the Wikipedia article:


  [Its existence] was affirmed by Friedberg and Muchnik in the 1950s using a novel technique called the priority method. They give a construction for a set that is simple (and thus non-recursive), but fails to compute the halting problem.