North Carolina Gov. Roy Cooper, now in the waning years of his second term, has suddenly found himself back on a campaign trail.

On Wednesday, flanked by supporters in a fifth-floor classroom at Cape Fear Community College in Wilmington, Mr. Cooper made a direct appeal to residents. But he was not looking for thousands of votes. Just one.

North Carolina’s Republican-dominated legislature has passed a bill banning most abortions after 12 weeks. Mr. Cooper, a Democrat, vetoed the bill. But to prevent the legislature from using its razor-thin supermajority to override his veto, Mr. Cooper is asking voters to pressure Republican lawmakers. Convincing just one legislator will keep the state’s current abortion law — allowing it up to 20 weeks — in place.

In Wilmington, he urged voters to send a message to their representatives in the legislature — “ask them to keep their promise” to preserve existing abortion laws, he said, referring to Republican lawmakers he said had previously signaled some support for abortion access.

Let them know, he said, “whether it’s a phone call, or it’s an email, or it’s a text.”

Mr. Cooper’s plea, and the showdown between the governor’s office and the legislature, represents an extraordinary moment in North Carolina politics, as well as in the nation’s volatile abortion fight.

Since the Supreme Court last year overturned Roe v. Wade, states have been free to severely restrict or ban the procedure, and many across the South have done exactly that. As a result, North Carolina has become an outlet for women in the region who could not get abortions in their home states.

For North Carolina, the override vote would be a consequential early test of the Republicans’ new, slim supermajority, since Tricia Cotham, a former Democrat, switched parties in April and voted in favor of the ban.

Override votes in the two chambers, each of which require a three-fifths vote of those present to succeed, have not yet been scheduled. But state lawmakers and lobbyists said over the weekend that they expected to see a vote as early as this week.

Republicans say the bill represents a compromise and is less restrictive than other bans that outlaw the procedure at conception or before most women even realize they are pregnant. Democrats say the bill is a disaster for women’s health, and erects all kinds of financial and logistical obstacles that would cut off abortion access for many women. They complained that Republicans rammed the initial votes through their chambers in two marathon sessions over 48 hours.

A Meredith poll in February showed that 57 percent of respondents supported the state’s current 20-week ban, or would expand it. Another 35 percent wanted the procedure restricted to 15 weeks or less.

Lauren Horsch, deputy chief of staff to Phil Berger, the Senate Republican leader, called the bill “a mainstream approach to limiting elective abortions in the second and third trimesters, supporting women and children, and ensuring that women have options available to them.” In a statement, Mr. Berger said he looked forward to “promptly overriding” the veto.

Mallory Finch, who came to Raleigh on Saturday to protest the governor’s veto, said, “North Carolina is a state for life, and there are people who want the act to go through.”

Democratic officials in districts across the state are trying to mobilize voters to oppose the bill. In New Hanover County, where Wilmington is located, party leaders organized a caller chain that contacted Republicans, including Ted Davis Jr., a Republican House member considered a swing vote, and Michael V. Lee, a Wilmington Republican state senator, every three minutes one day last week. Mr. Cooper believes both men might be movable on the issue.

Mr. Lee, however, said a 12-week restriction is in line with his thinking on abortion. In a text message, he said that Mr. Cooper has mischaracterized his position on the issue.

“I believe a woman should have the right to choose an abortion in the first trimester (3 months) with exceptions,” Mr. Lee wrote.

Mr. Davis has said in the past that he supported North Carolina’s current law. Mr. Cooper is also targeting the district of a fourth Republican, John Bradford, a House member who said shortly before his election last year that he had “no intention” of rolling back the 20-week law. Mr. Bradford did not respond to a request for comment.

The defection of Ms. Cotham, a former Charlotte-area educator who had served in the state legislature and made an unsuccessful run for Congress before returning to the North Carolina General Assembly this year, stunned Democrats.

In announcing her decision, she said she had been bullied by the party and was no longer aligned with them on some issues, including school choice.

“The modern-day Democratic Party has become unrecognizable to me and to so many others throughout this state and this country,” she said when she announced. “They have pushed me out.”

Ms. Cotham has historically been an outspoken supporter of abortion rights. When she was a Democrat, she accused Republicans of playing doctor. She also spoke publicly about her own harrowing experience with a lost pregnancy that required medical intervention. “This decision was up to me, my husband, my doctor and my God. It was not up to any of you in this chamber,” Ms. Cotham said in 2015. Still, she voted in favor of the 12-week ban after she switched parties.

Ms. Cotham did not respond to a request for comment.

On Thursday at the Modish Nail Spa in Mint Hill — the Charlotte suburb where Ms. Cotham lives — May Lopez said she was upset by the new abortion restriction.

“I feel terrible about it, because I think they’re just stripping the rights away from women. And I remember. I grew up in the days where my girlfriend died because of the hanger abortion and all that kind of stuff,” said Ms. Lopez, who votes largely for Democrats.

Frank McCullough, a Charlotte pastor, and his wife, Barbara McCullough, a retired schoolteacher, both voted for Ms. Cotham when she ran as a Democrat last year. Both said they felt betrayed by her decisions to switch parties and help Republicans pass more restrictions on abortion.

“I don’t believe in abortion, but I believe in the rights of a lady to make that choice between her and God,” Mr. McCullough said. “We voted for you and here you go turning your back on us.”

People who live in Ms. Cotham’s district said that while it leans Democratic, it also features a healthy presence of conservatives who back abortion restrictions.

On Wednesday afternoon in Wilmington, part of Mr. Davis’s district, swimmers at the Y.W.C.A. aquatic center were divided.

“I’m a Christian, and I believe that life begins at conception, and I’m against abortion altogether,” said Joyce Woodard, a retiree.

Emma Evans, a college student who was watching a swim lesson of the 4-year-old she was babysitting, said she was baffled by the passage of the abortion ban.

“I don’t know much about it at all, but I do know I’m for abortion” rights, she said. “A bunch of men are just making rules for these women’s bodies? It makes no sense to me.”

In an interview on Friday, Mr. Cooper appeared troubled by the political state of play. During more than six years in office, he had successfully vetoed more than 50 bills. The November midterms, which left Republicans just one vote shy of a supermajority in North Carolina, had threatened his control over the legislative process, which can be upended by a single lawmaker’s absence. Ms. Cotham’s party exodus last month deprived him of any remaining comfort.

“I knew things were precarious,” he said. “But then when Representative Cotham switched, and made it a supermajority by one vote in each chamber, we knew that it was going to be a much tougher fight.”

“I’m worried that women will die,” he said.

Motivating voters is no easy task: A number of people over the past week said they were only dimly aware of the fight, even if they felt strongly for or against abortion access.

Nick Decker was waiting for friends Thursday at the Crazy Pig, a barbecue joint in Mr. Bradford’s district. He said he was aware the governor had been in town that week “to try to sway some state legislators.”

“Charlotte and the metro area is very much a blue area,” he said. And he counted himself as a supporter of the governor and state Democrats.

He said he was not aware of the position of his Republican representative, Mr. Bradford. But, he added, “I’m very pro-choice.”

Bryan Anderson contributed reporting from Davidson, Charlotte and Raleigh, N.C.