Complex sentence: They began listening to the "sophisticated smoother modern jazz" of Dave Brubeck and the Modern Jazz Quartet "and became"... clothes obsessed, cool, (and) dedicated to R&B and their own dances.
Split into simpler sentences: They began listening to the "sophisticated smoother modern jazz" of Dave Brubeck and the Modern Jazz Quartet. "They became"... clothes obsessed, cool, (and) dedicated to R&B and their own dances.