Amid the frenzy backstage at the iHeartRadio Jingle Ball this month, an aura of businesslike calm seemed to follow Nick Jonas as he snaked his way through the underbelly of Madison Square Garden.

Wearing a heathered-burgundy sweater and tight black jeans, and with his immaculately parted hair and chiseled features, Mr. Jonas dutifully made the rounds required of a 22-year-old newly minted sex symbol.