Save for bad weather and scattered ballot shortages, voting appeared to run smoothly Tuesday in four state primaries that drew record crowds of Democrats eager to decide whether the first woman in history or the first black candidate would receive the party’s presidential nomination.

Heavy rain, sleet and ice forced at least 10 counties in Ohio to request permission to move, and a few polling spots were running 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, home to Cleveland.

Extraordinary interest in the Democratic contest between Obama and Hillary Rodham Clinton drew huge turnout in Texas, where both were running neck and neck. Long lines delayed the start of the state’s arcane precinct caucuses for an hour or more.

Those hearings cannot be held until every voter still in line when the polls close has cast a ballot. In some places, the lines were hundreds of people long.

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.

People trying to attend the Democratic caucus in the North Texas town of Little Elm waited in the cold for about two hours before being allowed inside. “It was extremely unorganized,” said Dan Perez, 30, a homebuilder.

In San Antonio, an attorney for the Mexican American Legal Defense and Educational Fund said it took an hour to sign up 126 Democrats who showed up to participate in the precinct caucus. Across the hall were five Republicans “sitting in a circle just looking at each other,” said Nina Perales.

“The turnout is extraordinary,” said Perales, who was at a grade school. “They had to put us in the gymnasium.” A fellow lawyer reported 300 Democrats at a nearby precinct, Perales said.

In Ohio, tallies were slow in coming from Cuyahoga County. Delays in that state have ranged from more than a month in the 2004 general election to five days in the 2006 primary, when absentee ballots had to be counted by hand. Especially worrisome to some advocates is Cuyahoga, the state’s most populous county, where election officials abruptly ordered the abandonment of electronic voting machines in favor of paper ballots for Tuesday’s primary.

“One of the things that is inescapable, is when you’ve got a paper system, it takes longer to count,” Chapin said. A record-setting 4 million voters were expected in Ohio, more than 50 percent of the state’s registration roll.

The delegate-rich states of Ohio and Texas are considered essential to Clinton, who’s lost the last 11 primaries to Obama, an Illinois senator who hopes to be the first black president.

Precincts in northern Sandusky County were also ordered to stay open an extra 90 minutes, this time by the secretary of state, after 300 to 400 voters were turned away when ballots ran out.

An ice storm struck Cleveland during rush hour, prompting nonprofit groups to stop roving election monitors.