A record number of voters overwhelmed Texas polling places, causing long lines and ballot shortages at some precincts and overflow crowds at postelection caucuses, while Ohio voters grappled with bad weather, power outages and ballot shortages.

Voting appeared to run smoothly in the two other state primaries Tuesday. The four contests drew large numbers of Democrats eager to decide whether the first woman in history or the first black candidate would receive the party’s presidential nomination.

“At this point in time, everybody’s looking for something different,” said Al Miller, a 20-year-old financial services worker who voted in Hurst, Texas. “I expected a lot of turnout because everybody’s trying to make a difference.”

Some 3.7 million Texans — 29 percent of the state’s registered voters — had cast ballots with about 85 percent of precincts reporting, said Scott Haywood, a spokesman for the Secretary of State. The state’s previous turnout record was 2.7 million, set in 1988.

Heavy rain, sleet, snow and ice forced at least 10 counties in Ohio to request permission to move, and a few polling spots ran on generators because of power outages. After hearing arguments from Barack Obama’s campaign, a federal judge ordered 21 precincts to remain open an extra 90 minutes after ballots gave out in Cuyahoga County, where Cleveland is located.

Washington County Elections Administrator Linda LaCross expected problems before the polls even opened.

“We don’t have enough ballots for the Republicans,” she told the Bryan-College Station Eagle. “They ordered 4,000 ballots, but we need more like 6,000 or 7,000. This is my worst nightmare.”

By midnight, up to 20 people were still in line at one Houston precinct, five hours after the cutoff. Harris County election administrator Johnnie German attributed the slowdown to the huge turnout and difficulty verifying whether some would-be voters were registered.

Similar scenes were reported elsewhere in Texas, where long lines delayed the start of the state’s arcane precinct caucuses for an hour or more. The hearings cannot be held until every voter still in line when the polls close has cast a ballot.

More than 2,000 people trying to attend the Democratic caucus in the northern Texas town of Little Elm waited in the parking lot for about two hours before being allowed into the building.

“It was extremely unorganized. It definitely caused a lot of frustration,” said Dan Perez, a 30-year-old homebuilder.

Election advocates worried that delays in Texas and Ohio could hold up final tallies for hours or days. Primaries held in Vermont and Rhode Island were counted relatively quickly.

Precincts in northern Sandusky County were also ordered to stay open an extra 90 minutes after 300 to 400 voters were turned away when ballots ran out.

Voters in Cuyahoga County seemed to adapt to paper ballots counted by optical scanners. In December, Secretary of State Jennifer Brunner ordered the county, the state’s most populous, to switch to paper ballots after touch-screen voting caused problems in 2006 and 2007.

With nearly 95 percent of precincts reporting early Wednesday, roughly 3.2 million of Ohio’s 7.8 million registered voters had cast a ballot. Excluding Cuyahoga and five other counties that could not yet report total results, the statewide turnout rate was 44.9 percent.

The state’s previous primary turnout record was 2.5 million, set in 2000.

Brunner initially predicted a 52 percent turnout, but said the weather, including an ice storm that struck Cleveland during rush hour, probably had an adverse effect.

“About the only thing we didn’t get today were locusts,” Brunner said.

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Associated Press writers Anabelle Garay in Dallas and Kathy Barks Hoffman in Cleveland contributed to this report.