Okay, there's one for today: The Defiant Ones
Clue 1: "The Fugitive Kind"
Clue 2: Sydney Poitier sings unassisted, "walking the line" referring to the character's feelings toward the other
Clue 3: Richard Brooks directed Sydney Poitier "too" in two films.
Using the clues and process of elimination through previous guesses in this thread (and prodded by wishful thinking), I'm thinking it might be Five Miles to Midnight (1962)?