SWANSEA, S.C. — Valerie Biden Owens traveled past swamps, harvested fields and parked tractors to a small church, where she stepped off a bus to address voters. She had her talking points memorized.

Just as she had done at every other stop campaigning for her brother, former Vice President Joseph R. Biden Jr., Ms. Owens invoked the deep personal suffering he endured, and how it infused him with empathy and toughened him against adversity.