One morning this week, Eric Adams sat down at a sidewalk table outside the Washington Square Diner, in the West Village. Two decades ago, at the end of his career in the N.Y.P.D., Adams had worked nearby, in the Sixth Precinct. “This was my post,” he said. A waiter plopped a stack of thick menus on the table. Adams, who wore a crisp white dress shirt, with cufflinks, credits a strict vegan diet and exercise regimen with reversing a diabetes diagnosis. He ordered a peppermint tea.