Kamala Harris’s razor-sharp debate skills have powered her political ascent. Her success next week will turn on how well she adapts to an unpredictable rival. Few of the people gathered in the San Francisco church knew quite what to expect from Kamala Harris. It was 2003, and the two men she faced in her first political campaign were well-known brawlers locked in a rematch to become the city’s district attorney — one of whom was her former boss. Ms. Harris was a little-known government lawyer who had landed in the local gossip pages for dating Willie Brown, one of the...