HOUSTON — Senator Obama is seeking a knockout against Senator Clinton today as voters turn out in large numbers in Texas and Ohio, where the former first lady desperately needs victories to salvage her once-powerful candidacy for the Democratic presidential nomination.

Both expressed confidence in their chances, but their campaign teams also acknowledged that split decisions and close votes could prolong the battle for at least another month — or more.

“I feel really good about today,” Mrs. Clinton told reporters after a visit to a polling place at a Houston elementary school in a largely Hispanic neighborhood today. “Let’s wait and see what the voters have actually decided — I think it’s going to turn out well.”

Mrs. Clinton planned to stop at another polling place in Dallas before heading to Ohio for more campaign events. She will await results in Columbus before returning to Washington tonight.

Mr. Obama began the day at the Houston Livestock Show and Rodeo, where he shook hands with people at a Future Farmers of America exhibit, posed for pictures, and climbed onto a tractor.

“I hope we do well, but we’re working hard,” he told reporters before heading to San Antonio, where he planned to await returns.

Polls show tight races in both Texas and Ohio. The Obama campaign saw Texas as their best opportunity, while the Clinton campaign saw Ohio as theirs. Texas offers 228 delegates, Ohio 164.

Voters also went to the polls today in Rhode Island, which offers 21 delegates, and Vermont, 15.

In Ohio, voting sites were busy despite heavy rain across the state.

Eric Gingerich, 36, a junior high social studies teacher, normally votes Republican, but voted today for Mr. Obama at an elementary school in Hilliard, Ohio, near Columbus.

“I like how he can bring the two parties together and the country together,” he said.

Republican scandals in Ohio have made him more open to Democrats, but Mr. Gingerich said Mrs. Clinton is too polarizing.

“If anybody is going to pull me over to the Democrats it’s Barack, not Hillary,” he said.

The economy was on the mind of Gretchen Genung, 61, a state employee who voted for Mrs. Clinton in Cincinnati.

“We know where the economy is right now, we know where it’s going and we need somebody in there like Hillary who has the experience,” Ms. Genung said.

In Providence, R.I., Sharon Carpentier, 46, said she voted for Mrs. Clinton because she wants to see a female president in her lifetime and because she admires her perseverance through the public airing of her personal problems.

“If she can withstand that kind of heat, she can withstand a lot of things,” she said.

A 34-year-old bicycle frame builder, Brian Chapman, also voted in Providence — but for Mr. Obama, whom he compared to President Kennedy.

“He’s evoking the feeling for me of positivity, he’s making me actually proud to be an American and proud to be living in this country instead of being ashamed as I am right now under the current political leadership,” Mr. Chapman said.

Mr. Obama was dogged in the days leading up to today’s primary by allegations that he had overstated his opposition to the North American Free Trade Agreement to win votes in Ohio, and his ties to a Chicago businessman, Antoin Rezko, on the day that jury selection began in the political corruption trial of the real estate developer and fast-food magnate.

“Tony Rezko was a friend and supporter of mine for many years. These charges are completely unrelated to me, and nobody disputes that,” Mr. Obama said at a news conference yesterday in San Antonio.

Mrs. Clinton has sounded a populist economic theme leading up to today’s primaries as she courted voters who have suffered with the decline of manufacturing in the industrial Midwest and Ohio. Then, in military-friendly Texas, Mrs. Clinton broadened her theme to include veterans’ issues and to trumpet her backing from a string of top military officers.

Mrs. Clinton has also worked to underscore her core campaign theme that she’s the more experienced on the issue. She held a one-hour town hall meeting where she strode the stage surrounded by a friendly audience and took questions selected from the thousands that were submitted on issues ranging from health care to education to veterans issues. Her campaign purchased time on a sports-oriented cable network to broadcast the event around the state, and the event was streamed on the campaign’s Web site.

Lucy Salazar, who voted at the Houston elmentary school Mrs. Clinton visited today, predicted the former first lady would do well in Texas, noting she has paid particular attention to the state’s Hispanic community.

“We’ve not had a president come to the neighborhood like this ever before,” Ms. Salazar said.

Despite an exchange of jabs between the two candidates over NAFTA and who would be better able to respond to a national security crisis, Mrs. Clinton called the Democratic race “one of the most civil and positive primary campaigns I can remember.”

An Obama campaign manager, David Plouffe, called today “the last big window of opportunity” for Mrs. Clinton, noting that “enormous leads” she enjoyed as recently as two weeks ago had dwindled or evaporated.

Still, he said Mr. Obama was mindful that “this could go on for some time. We’re prepared for whatever situation occurs.”

The Republican presidential contenders, Senator McCain and Michael Huckabee, also campaigned in Texas, though voter interest centered on the closer Democratic race.

“I understand the interest in very close races,” Mr. McCain said of the close Democratic contest. “As you know, there was a lot more interest earlier in our Republican primaries.”

Mr. McCain, the Republican nominee-in-waiting, began the day by rallying about 200 supporters crammed into a Mexican restaurant and bakery in San Antonio. He planned to hold a town hall-style event in Houston and watch election returns later in Dallas.

The Arizona senator criticized the Democrats for their desire to set a timetable for withdrawing American troops from Iraq. “Senator Obama and Senator Clinton want to get out of Iraq as fast as we can,” he said as the crowd booed. “They say they want to set a date for withdrawal. I tell you, if they did that, my friends, we would be back.”

A man shouted: “Let’s win the war!”

“Exactly,” Mr. McCain replied.