The former first lady has lost 11 straight primaries and caucuses, and trails her rival in convention delegates. Obama has won a pair of big union endorsements in the past two days.
In a university auditorium in the heart of Texas, the two rivals agreed that high-tech surveillance measures are preferable to construction of a fence to curtail illegal immigration.
They disagreed on the proper response to a change in government in Cuba in the wake of Fidel Castro's resignation. Clinton said she would refuse to sit down with incoming President Raul Castro until he implements political and economic reforms. Obama said he would meet "without preconditions," but added the U.S. agenda for such a session would include human rights in the Communist island nation.
They also sparred frequently about health care, a core issue of the campaign.
Clinton said repeatedly that Obama's plan would leave 15 million Americans uncovered.
But he, in turn, accused the former first lady of mishandling the issue by working in secrecy when her husband was in the White House.
"I'm going to do things differently," he said. "We can have great plans, but if we don't change how the politics is working in Washington, then neither of our plans are going to happen."
Clinton was combative and complimentary by turns, and reflected on her well-known personal struggles in the debate's final moments.
"Everyone here knows I've lived through some crises and some challenging moments in my life," she said - a thinly veiled but clear reference to her husband's affair with Monica Lewinsky and subsequent impeachment. But she added that nothing she had been through matched the everyday struggles of voters.
Then, offering unprompted praise to her rival, the one-time front-runner said, "No matter what happens in this contest, I am honored to be here with Barack Obama."
Both candidates were plainly popular with the debate audience. During one break someone in the crowd shouted "Si, se puede," Spanish for Obama's trademark phrase, "Yes we can."
Clinton largely sidestepped a question about so-called superdelegates, members of Congress, governors and party leaders who were not picked in primaries and caucuses. She said the issue would sort itself out, and "we'll have a united Democratic party" for the fall campaign.
But Obama, who has won more primaries and caucuses said the contests must "count for something ... that the will of the voters ... is what ultimately will determine who our next nominee is going to be."
Clinton went into the debate needing a change in the course of the campaign, and waited patiently for an opening to try to diminish her rival, seated inches away on the stage. "I think you can tell from the first 45 minutes Senator Obama and I have a lot in common," she said.
Barely pausing for breath, she went on to say there were differences.
First, she said she had seen a supporter of Obama interviewed on television recently, and unable to name a single accomplishment the Illinois senator had on his record.
"Words are important and words matter but actions speak louder than words," she said.
Obama agreed with that, then noted that Clinton lately had been urging voters to turn against him by saying, "let's get real."
"And the implication is that the people who've been voting for me or are involved in my campaign are somehow delusional," Obama said.
Clinton also raised Obama's use in his campaign speeches of words first uttered by his friend, Massachusetts Gov. Deval Patrick.
"If your candidacy is going to be about words then they should be your own words," she said. "...Lifting whole passages from someone else's speeches is not change you can believe in, it's change you can Xerox."
The debate audience booed.
Obama said the entire controversy was evidence of a "silly season" that the public finds dispiriting. Besides, he said of his speeches at one point, "I've got to admit, some of them are pretty good."
The two rivals sat next to one another in swivel chairs in a University of Texas auditorium for the 90-minute debate, one in a dwindling number of opportunities for the former first lady to chart a new course in the presidential race.
She has lost 11 straight primaries and caucuses to Obama - including an overseas competition for support among Americans living aboard - and has fallen behind in the chase for the number of delegates needed to become the presidential nominee.
Obama's strong showing has made him the man to beat in a historic struggle between a black man and a white woman, and even former President Bill Clinton has said his wife must win both Ohio and Texas early next month to preserve her candidacy. New polls show Texas a dead heat, and give Clinton a lead in Ohio, but far smaller than the one she held in recent weeks.
Rhode Island and Vermont also vote on March 4, but offer far fewer delegates and have drawn less attention.
The encounter was the 19th in an episodic series of debates and forums, a run that has ranged from highly civilized to hotly confrontational.
The last time the two met, in Los Angeles, they sat side by side and disagreed politely. But in an earlier encounter last month, in Myrtle Beach, S.C., each accused the other of repeatedly and deliberately distorting the truth for political gain in a highly personal, finger-wagging showdown.
In The Associated Press' delegate count Thursday, Obama had 1,358.5 to 1,264 for Clinton. It takes 2,025 delegates to claim the nomination at this summer's convention.
In a further sign of his growing strength, Obama won the endorsement during the day of the Change to Win labor federation, which claims 6 million members. The Teamsters union announced its support for Obama on Wednesday.
The debate was sponsored by CNN, Univision and the Texas Democratic Party.
