LANCASTER, Pennsylvania — “This is my kind of place,” said Vice President Mike Pence as he looked around Lyndon Diner, its booths packed with supporters and its aisles filled with waitstaff in T-shirts emblazoned with shamrock logos.

While Washington obsessed over impeachment and waited for the Senate’s final vote, Pence got out of town to woo voters in Pennsylvania, dispensing folksy charm and selfies in a crucial battleground state.