Opponents of a controversial amendment to ban gay marriage, civil unions and domestic partnerships in North Carolina were engaged in last-ditch attempts to get their supporters out on Tuesday, as the latest polls show it is likely to pass.

Fiercely-fought campaigns, with television adverts from both sides of the debate, ensured a record turnout during the state's early voting period. High-profile backers have been deployed on opposing sides in recent days.

Former president Bill Clinton has urged people to vote against Amendment 1 in recorded telephone calls, while evangelist Billy Graham, who has a stretch of road named after him in North Carolina's largest city, Charlotte, has endorsed the amendment in full-page newspaper ads. The latest polls have put support for the amendment at 55%, against 39% for those who oppose it.

All that's left is a few hours of last-minute phone and door-to-door calls by volunteers.

Despite the odds, Jeremy Kennedy, of the anti-amendment group Coalition to Protect All NC Families, insists that there is still a chance the amendment may be defeated.

He said: "The name of the game is getting people out to vote. We've done TV commercials, we've done the mailings, we've been paving the way for this for months. Now it's about making sure we talk to the people who are against the amendment and getting them to the polls."

Kennedy dismissed the recent polls and said that between now and the close of voting at 7.30pm, "anything could happen." He said that pollsters often rely on people who have voted in recent primaries, who tend to give accurate predictions of the result. But if enough people turn up who have never voted in a primary, there may be an upset. He is banking on a strong turnout from "unlikely voters" drawn to the polls by their desire to vote against the amendment.

"We've been contacting voters for a long time. Our analysis shows young people are voting, as are unaffiliated voters," he said. Both groups tend to support same sex unions and are likely to vote against the amendment.

Same-sex marriage is already illegal in socially conservative North Carolina, the last of the former confederate states not to have a state constitutional ban. The main group in favour, Vote for Marriage NC, has enlisted massive support via the clergy in a state with a strong evangelical vote.

Signs display messages about gay marriage in front of the Devon Park United Methodist Church polling site in Wilmington, North Carolina. Photograph: Ken Blevins/AP

Vote for Marriage NC did not respond to a request to comment for this article. But a statement by over 100 pastors and religious leaders posted on its website urged voters to vote for the amendment. It said: "Across the United States of America, marriage has been under attack. In states where same-sex 'marriage' has been imposed on society, the religious freedoms of those who support marriage between one man and one woman have been threatened.

"We need a constitutional amendment preserving marriage so that our religious freedom is not endangered by those who seek to make marriage a genderless institution."

Kennedy stressed that his campaign has a list of over 400 church leaders and that over 100 African American clergy have come out against the amendment, but concedes that his opponents have the white evangelical congregations.

"They have made this a moral thing – 'God wants you to go out and vote'," he said. "It's hard to argue against that. But we are not asking people to make it a moral issue about whether gay marriage is right or wrong. Make it a decision about whether this amendment is right or wrong."

If passed, Amendment 1 would alter the state constitution to declare "marriage between one man and one woman" as "the only domestic legal union that shall be valid or recognized."

In other words, it would not only prevent same-sex marriage but ban any state recognition of civil unions and domestic partnerships.

Campaigners against Amendment 1 have concentrated their arguments on the wider implications of the law, and its effect on those for whom it was never intended to affect, like the children of those in domestic partnerships or to lessen the protection given to victims of domestic violence who are unmarried.

"This amendment can hurt people. What will change is domestic partnership, offered in eight parts of the state, will be illegal in North Carolina, and children will lose health benefits. You can disagree with same-sex marriage but still be against the amendment," said Kennedy.

Holning Lau, an associate professor of law at the University of North Carolina, who has written extensively on the implications of Amendment 1, said: "The language is very broad compared to other states. The wording says that heterosexual marriage is the only domestic legal union that the state can recognise. It is a common misconception that it would only affect same-sex marriage."

Lau said there were already parts of North Carolina that recognise domestic partnerships for public employees in terms of health benefits and also benefits for students in the public university system. These health schemes would have to be dismantled if the amendment passes, he said, because public employees will be banned from offering benefits to domestic partnerships.

A report by Lau and others concluded it was "impossible to predict" how courts would resolve issues such as protection for victims of domestic violence, raised by the amendment's vague language.

It concluded that it would take years of expensive litigation to settle its meaning and "when the dust clears, [all] unmarried couples would have fewer rights over their most important life decisions than they would have had otherwise."

In a blog post, Tom Jensen, of Public Policy Polling, said that in, some sense, "North Carolinians are voting against their own beliefs."

He wrote: "53% of voters in the state support either gay marriage or civil unions, yet a majority also support the amendment that would ban both."

"The reason for that disconnect is even with just 24 hours until election day only 46% of voters realize the proposal bans both gay marriage and civil unions. Those informed voters oppose the amendment by a 61-37 margin but there may not be enough time left to get the rest of the electorate up to speed."