BELFAST, Maine—The streets of downtown Belfast were charged with energy: politics was working its way through the public square. On one corner, older men and women held LGBT rainbow flags. The June 6 Pride parade would soon weave through the 19th-century brick buildings that anchor the small coastal city. Not far away, a red-haired, bearded man, flanked by a staffer, said goodbye to his supporters. The two men joked around as they ambled to a nearby pickup. Graham Platner, Democratic Senate hopeful, and the man in the eye of a media storm, had been offering last-minute encouragement to his volunteers in the final stretch before Maine’s June 9 primary. He had come from a similar event in Ellsworth—the location of his campaign headquarters, situated in another very progressive downtown not far from his home base in Sullivan....