RALEIGH, N.C. — With swift and brutal efficiency, North Carolina Republicans welcomed an incoming Democratic governor this month to an elite but hardly desirable club: the handful of governors around the nation who face veto-proof majorities of the opposing party.

First, the state legislature stripped away some significant powers from Gov.-elect Roy Cooper before he takes office on New Year’s Day. Then, last week, lawmakers backed away from a compromise Mr. Cooper thought he had brokered to repeal the state’s notorious “bathroom bill” limiting protections of gay rights, which has exposed North Carolina to costly boycotts. “There is rancor between this governor-elect and this legislature,” said Courtney A. Crowder, who was a legislative liaison for an earlier Democratic governor, Beverly Perdue.

Still, it is not yet clear that Mr. Cooper is condemned to four years of disdain in which Republicans in the General Assembly pass laws he is powerless to stop, while ensuring that his own initiatives land with the thud of a doorstop.

Stymied by hostile lawmakers, he has other options for advancing his priorities, if the experiences of fellow members of his new club, both Democrats and Republicans in other states, are a guide.