OLYMPIA, Wash. — The Republicans picked up a governor’s office Tuesday that has been in Democratic Party control, in North Carolina, adding to a streak of gains in recent elections that has given the party a strong majority in state capitals.

But Democratic incumbents held on in Delaware and Vermont in an election year when many voters were distracted from duels for governor by other races and issues — from the presidential race to the control of Congress. And Democrats also held onto the governor’s seat in New Hampshire, where a popular Democratic incumbent chose not to run again.