MOULTONBOROUGH, N.H (Reuters) - Republican White House hopeful Michele Bachmann, beset by falling poll numbers, staff departures and reports of fundraising troubles, on Sunday vowed to continue her campaign through the New Hampshire primary, which is likely to be in early January.

Stumping in the early primary state for the first time since making her candidacy official in June, Bachmann kept up her attacks on President Barack Obama and urged Republicans not to compromise with a moderate nominee to oppose him.

A new poll shows the Minnesota Congresswoman, a Tea Party favorite, tied for tenth in New Hampshire after peaking as high as second in June.

Bachmann on Sunday mixed harsh rhetoric with humor as she campaigned in a state she has largely ignored in favor of Iowa, which is seen as more receptive to her conservative message.

"I want to be your sweetheart here in New Hampshire," she told a meeting of about 50 Tea Party supporters.

"The thing that I would look forward to more than anything as the Republican nominee is taking Barack Obama on in the debates and holding him accountable for four years of destroying the country."

Bachmann also took a veiled swipe at Republican frontrunner Mitt Romney as she campaigned in Moultonborough, on the shores of Lake Winnipesaukee, about 20 miles from Romney's vacation home in Wolfeboro.

Polls since 2009 have given former Massachusetts governor Romney a wide lead in the state among Republican hopefuls.

"It's not good enough to settle for anyone but Barack Obama," she said. "This is the election where we have to have a constitutional conservative."

Still, Bachmann, an outspoken evangelical Christian and social conservative, came to the defense of Romney's Mormon faith, which was criticized by an evangelical pastor at a conservative conference in Washington on Friday.

Texas pastor Robert Jeffress, who introduced and endorsed Texas Governor Rick Perry at the Value Voters meeting, told voters to choose a candidate other than Romney, and later told reporters that Mormonism was a "cult."

"We don't have a test for people when they go into the White House," said Bachmann, who often espouses her evangelical beliefs while campaigning. "We do believe in tolerance and liberty for all Americans."

Bachmann said the long summer debate over raising the federal government's borrowing limit had kept her away from New Hampshire. "I wish I could have come back earlier but I was fighting the debt ceiling all summer," she said.

Bachmann's poll numbers peaked over the summer in the run-up to her victory in the Iowa straw poll, but have been falling since Perry's entry into the race in August and most recently with the rise of pizza magnate Herman Cain.

Since September her campaign manager and pollster have resigned and a number of Iowa campaign staff have returned to her congressional office staff.

Reports suggest her campaign raised less than $5 million in the third quarter of the year, well below the levels raised by Romney and Perry.

The bad news has dented the confidence of even her core supporters.

"I'm not sure Michele has a great chance," said Don Ewing, a Tea Party supporter from Meredith, N.H., who said he was still deciding whether to support Bachmann or Cain. "You have to support the people you think will do the best job."

(Reporting by Jason McLure. Editing by Ros Krasny and Peter Bohan)

Photo by Gage Skidmore.

Mochila insert follows.