President-elect Barack Obama’s Congressional outreach effort appears to be having the desired effect on Capitol Hill, with even the most skeptical of Republicans feeling the love from the man who has made “changing the way Washington does business” his mantra.

But the new comity has not prevented family spats from breaking out between Obama and Democrats — and Republicans say they are determined to assert themselves against the popular new president, even as some acknowledge it won’t be easy at first.

Sen. Tom Coburn (R-Okla), a conservative whose bitter struggle with Majority Leader Harry Reid (D-Nev.) over public lands legislation entangled the Senate for months, says Obama’s courtship is genuine.

“I think he is sincere — all the time,” Coburn said. “I believe it because I know him.”

Sen. Ben Nelson (D-Neb.) rhapsodized Tuesday about the new era of consultation.

“I don’t know how it could be any better,” he said.

But the warm feelings could also cause trouble for Democratic leaders. Republicans are signaling they intend to invoke Obama’s promises of bipartisanship when Democrats start to play rough.

“He and his staff have done a good job of reaching out to us,” Sen. Lamar Alexander (R-Tenn.) said. “I hope his tone is reflected in the Senate Majority Leader’s tone.”

And all the happy talk has resulted in at least some miscommunication. Reid at one point was steamed enough to exclaim that he doesn’t work for Obama. And the president-elect managed to nominate a new CIA director, Leon Panetta, without consulting Sen. Dianne Feinstein (D-Calif.), the new chairman of the Intelligence Committee.

Obama has been to Capitol Hill three times to meet with Democrats and Republicans since arriving in Washington just over a week ago. President George W. Bush was almost never to be seen in the halls of Congress, unless it was to deliver a State of the Union speech, to attend a St. Patrick’s Day lunch or to be inaugurated.

Obama’s trips to Capitol Hill have been supplemented with phone calls and significant negotiations between Obama and Congressional staff as Congress seeks to move legislation quickly to boost the economy.

Several lawmakers, including some Republicans, compared Obama’s style favorably to that of Bush, known around Capitol Hill for a “my way or the highway” approach.

“This seems like it will be an administration that is willing work with you, as opposed to the last administration that was willing to work on you,” Nelson said.

Sen. Olympia Snowe (R-Maine), a moderate and a longtime thorn in the side of the Bush White House, remarked that there appears to be an unmistakable difference between Obama’s approach to Congress and that of Bush.

“It is a very different effort, frankly,” she said. Obama is “reaching out early on and setting the tone from the start by which, optimally, he will govern.”

She said she has been assured by Vice President-elect Joseph Biden that “there will be follow-up” when input from Capitol Hill is sought.

Several Senators, including Snowe and Nelson, suggested Obama’s approach will help him move votes.

Indeed, Democrats, some of whom walked into Tuesday’s caucus lunch with Obama demanding detailed written assurances about the use of the second tranche of the $700 billion Troubled Assets Relief Program, emerged from the gathering feeling better about it.

Obama wants the money as soon as possible, and he appears likely to get it in what would be his first significant legislative victory — even though some Democrats fear voter skepticism and anger about the size of the bailout package.

Some Republicans are concerned that, whatever Obama’s sincerity, they risk being co-opted into supporting legislation that will nevertheless be credited to Democrats and Obama when it passes — while they will share the blame if voters hate it.

“Everyone wants him to be successful,” Sen. Jeff Sessions (R-Ala.) said. But if Republicans buy into something they shouldn’t, he said, “it’s more difficult to assert that we’re moving on a dangerous road and Obama’s leading it.”

Sen. John Cornyn (Texas), the new chairman of the National Republican Senatorial Committee, was careful to compliment Obama for his overtures, but he added that Republicans would find out “soon enough” how sincere Obama is.

And either way, he said, “I’m confident that there will be sufficient differences to distinguish Republicans and Democrats as we move forward.”