In Nevada, again raised the topic of rival lack of management experience, showing that their earlier truce on racial matters did not extend to other issues in a tight Democratic race. Polling in the state shows the two, and main rival , in a statistical dead heat.
Republicans have had three winners in their first three major contests. , who won in New Hampshire, has led polls in South Carolina. But - a Baptist preacher and former Arkansas governor - could get a boost from Christian evangelicals whose support helped him win in Iowa. And Romney has fresh momentum after his Michigan victory.
The latest polls show a close Republican race in South Carolina.
The state's Republican governor Mark Sanford says GOP voters are frustrated and picky this year, reports CBS News correspondent Bill Whitaker.
"People are really going through the candidates with a fine tooth comb," Sanford said.
"It's going to be very interesting in South Carolina," CBS News chief Washington correspondent Bob Schieffer said. "Huckabee will have a constituency… McCain will have a constituency … And Thompson… he's going to get some votes."
"We still may be after South Carolina saying the same thing we're saying tonight - that it's still anybody's guess who's going to get the nomination," Schieffer adds.
Romney, a former Massachusetts governor, took the economic message that helped him in Michigan to South Carolina on Wednesday, swapping talk of resurrecting the auto industry with a pledge to pay attention to textile and other industrial job losses that have punished the South.
"I'm not willing to declare defeat on any industry where we can be competitive. I'm going to fight for every job," Romney said.
Romney, a millionaire businessman and former Massachusetts governor, captured his native Midwestern state of Michigan on Tuesday by offering hope to revive its beleaguered industrial economy.
Romney had 39 percent of the vote in Michigan. McCain had 30 percent and Huckabee 16 percent. No other Republican fared better than single digits.
McCain has been the latest front-runner in national polls, but leads have been changing frequently. He is banking in South Carolina on his three rivals splitting the far-right Republican vote and giving him an avenue among moderate Republicans who have moved into the traditionally conservative state in the last eight years.
McCain, a veteran U.S. senator and former prisoner of war in Vietnam, hopes for a strong showing thanks in part to his military background in a state with many military veterans.
He promised 100 supporters in Greenville on Wednesday that he would ensure better health care for active duty and retired service members.
McCain spoke in a state with a large population of veterans and reservists, including 1,500 National Guard troops serving in Afghanistan. McCain declared "we are succeeding in Iraq" and would not be if Democrats had their way.
"Different people are winning these different major contests and I think a different person will win Saturday in South Carolina. No one has settled in on anyone," Thompson said.
While other Republicans had their sights set on South Carolina, former New York City mayor Rudy Giuliani was campaigning in Florida, which votes Jan. 29, and where he hopes to realize his first win.
A victory in Florida would boost Giuliani past his opponents when more than 20 states, including California and his home state, vote on Feb. 5. He was counting on support from New Yorkers who retired in Florida.
In the Democratic field, Clinton, who won in New Hampshire; Obama, who secured a win in Iowa; and Edwards, a former North Carolina senator and vice presidential candidate, were locked in a tight race going into Nevada's caucuses, also on Saturday.
In Las Vegas on Wednesday, Clinton seized on Obama's acknowledged lack of management experience. She said she was taken aback by comments Obama made in a previous night's debate suggesting it was more important for a president to bring vision to the job than to make sure "paperwork is being shuffled effectively."
Clinton, who wants to be the U.S. first female president, and Obama, his sights set on becoming the country's first black president, were coming off a toned-down television debate Tuesday night in which they pledged to tamp down arguments between their camps over race.
The dispute focused on comments interpreted by Obama's camp as belittling slain civil rights leader Dr. Martin Luther King Jr.'s contributions to the movement, and raised fears among Democrats that infighting could damage the party's prospects for capturing the White House in the November general election.
Race, however, is expected to factor into the Democratic campaign, and some Clinton allies were already betraying worries about the impact of minority voters on her campaign in the state.
The concern was that thousands of Culinary Workers local members in Las Vegas, whose union has endorsed Obama, will attend caucuses conveniently located near their work places and hand him a victory. The union represents 60,000 employees, about 45 percent of them Latino, 30 percent white and 10 percent black.
It could also factor in South Carolina, where the South's first nomination contest is upcoming. Blacks are likely to account for at least half the electorate in South Carolina.
The state's most prominent black politician, Democratic Rep. Jim Clyburn, has not endorsed any of the contenders. But with his constituents likely to back Obama, he let his veil of neutrality slip recently with comments critical of Clinton.