GLEN DAVID ANDREWS hopped down from an outdoor stage at the New Orleans Jazz & Heritage Festival in May, leaving his trombone behind. He sang in a powerful raspy voice, inflected with just a hint of Louis Armstrong. Segueing from one song to another - the controversial 1920s classic “Black and Blue” to the more recent brass-band tune “Cell Block Nine,” for example - he sprinkled each with improvised lyrics. “It's my time,” he shouted between numbers.

Andrews, 28, has a lanky 6-foot-4-i...