Bluesman Bobby Rush is living high off the hog, or at least that’s how the down home, Louisiana native might describe it. “We talk about chickens, donkeys, monkeys and alligators—anything that moves on the ground,” Rush once laughingly proclaimed.

At age 83, the guitarist, vocalist, harmonica player and composer, whose first gold record was 1971’s “Chicken Heads,” is celebrating having been honored with his first Grammy for his deeply soulful, often funky and funny and always straight-up h...