A new set of polls conducted in early-voting states shows a tightening race among both Republican and Democratic presidential candidates, particularly in Iowa, which holds its caucuses Jan. 3.

A CBS/New York Times poll of 1,273 likely Iowa voters conducted Nov. 2-11 showed the frontrunners in both parties feeling heat from their competitors.

In the GOP race, the poll showed former Arkansas Gov. Mike Huckabee closing in on leader Mitt Romney. Huckabee earned 21 percent while Romney earned 27 percent. Former New York City Mayor Rudy Giuliani, who is leading in most national polls, clocked in at 15 percent.

In the Democratic race, Illinois Sen. Barack Obama and former North Carolina Sen. John Edwards were shown in a near tie — with 22 and 23 percent respectively — and catching up to Hillary Clinton, who polled at 25 percent.

Another Iowa poll of 600 likely voters from each party conducted by Strategic Vision, LLC from Nov. 9-11 showed similar standings.

In the GOP race, Romney led with 30 percent, followed closely by Huckabee with 19 percent. Giuliani earned just 12 percent.

The poll showed Clinton with 29 percent on the Democratic side, but Obama nipping at her heels with 27 percent. Edwards was trailing with 20 percent in that poll.

Another CBS/New York Times poll of 719 likely New Hampshire primary voters conducted from Nov. 9-12 showed the state's frontrunners carrying stronger leads.

The poll had both Giuliani and Arizona Sen. John McCain with 16 percent, but far behind Romney, with 34 percent.

Clinton, too, held a stronger lead in the New Hampshire poll, with 37 percent. Obama earned 22 percent in the survey, while Edwards, with 9 percent, trailed far behind the 18 percent of undecided voters.

The polls in Iowa and New Hampshire show a political landscape much different than that reflected in national surveys, where Clinton has cleared an immense gap between herself and Obama, and where the GOP horserace looks like more of a sure thing. National polls showing Giuliani followed by former Tennessee Sen. Fred Thompson with Romney and Huckabee far in the distance look nothing like the races in Iowa and New Hampshire, where Romney and Huckabee are poised to shake up the frontrunners nationally.

But both CBS polls reflected a common thread among voters when it comes to concerns about trust and pandering with Clinton. Thirty-eight percent of New Hampshire voters said Clinton only says what she thinks people want to hear. That number was 18 percent for Obama and 29 percent for Edwards. In Iowa, 48 percent of voters said Clinton says what she thinks people want to hear. Again, that number was lower among her opponents — 12 percent for Obama and 17 percent for Edwards.

However, the New Hampshire poll reveals Clinton's husband could help her on the campaign trail. The poll showed 41 percent of voters saw the former president as an asset that would make them more likely to vote for Hillary Clinton. Fifty-two percent said it makes no difference.

Finally, a SurveyUSA poll taken of South Carolina voters from Nov. 9-11 had Giuliani with 26 percent in the GOP race, followed by Romney with 20 percent and Thompson with 18 percent.

The poll, taken of 577 likely Republican primary voters and 447 likely Democratic primary voters, showed Clinton leading with 47 percent in the Democratic race, followed by Obama with 33 percent and Edwards with 10 percent.