A new poll has former Vice President Joe Biden leading Beto O'Rourke in the Texas presidential primary and locked in a close race one-on-one with Donald Trump.

The survey, conducted by Emerson College for The Dallas Morning News, had a 3 percentage-point margin of error, and Biden and Vermont Sen. Bernie Sanders each led Trump by only 2 percentage points.

The poll also signals that even with two favorite sons in the race, O'Rourke and Julián Castro, Lone Star State Democrats want a familiar face as their nominee.

O'Rourke is trailing Biden in the poll, perhaps, because a clear majority of Texas Democrats would rather see him run for Senate against Republican incumbent John Cornyn.

Sanders, the 2016 runner-up to Hillary Clinton for the party's nomination, was third.

The poll was conducted Aug. 1-3 among a sample of 1,033 registered voters with a margin of error of plus or minus 3 percentage points. For the Democratic presidential primary, about 400 people were surveyed with a margin of error of plus or minus 4.9 percentage points.

The poll also projects a wide-open Democratic primary race for the Senate seat held by Cornyn, the longtime incumbent. At 19%, "someone else" is leading the field, a blow to former Air Force helicopter pilot MJ Hegar, who's been campaigning for most of the year.

That "someone else" is leading the entire field is an oddity, but reflects the complexity of the primary race and the conundrum felt by many Democrats.

Hegar was the choice of 10% of those polled, followed by state Sen. Royce West at 8%, former U.S. Rep. Chris Bell at 7% and Houston City Council member Amanda Edwards at 5%. A whopping 51% of respondents were unsure.

West, Bell and Edwards are all relatively new to the race.

"I wouldn't be surprised to see other people jump into the race," said Spencer Kimball, the Emerson College polling director. "It's just that wide open."

The news is not great for Cornyn, the powerful incumbent who's held the seat since 2003. Only 37% approved of his job performance, while 31% disapproved. The polls found that 33% of Texans were neutral or had no opinion.

"That's not a good place to be. It shows that he's vulnerable," Kimball said of Cornyn's poll numbers, noting that Texas Gov. Greg Abbott had an approval rating of 50%, with only 27% not liking his performance.

"The good news for him is that he doesn't have disapproval in the GOP," Kimball said, noting his rating was 57% approval and 16% disapproval among Republicans.

Battleground Texas

Texas could be the biggest battleground in the country, as Democrats try to win a statewide contest for the first time since 1994.

In 2018, O'Rourke emerged as the party's savior, staging a dramatic campaign that came within 2.6 percentage points of beating Republican Sen. Ted Cruz.

Democrats had hoped O'Rourke would run for Senate against Cornyn, and it shows in the Emerson poll numbers.

Forty-six percent of voters said O'Rourke should drop out of the presidential race and run for Senate, while 25% said he should keep running for president and 29% weren't sure. About 51% of Democrats said O'Rourke should run for Senate instead of the White House.

Democratic presidential candidate and former Texas Rep. Beto O'Rourke speaks during a vigil for victims of Saturday's mass shooting at a shopping complex Sunday, Aug. 4, 2019, in El Paso, Texas. (AP Photo/John Locher) (John Locher / AP)

Before the mass shooting in El Paso caused O'Rourke to halt his campaign, he had been campaigning on his ability to win Texas' 38 electoral votes and defeat Trump for the White House.

O'Rourke also touted a University of Texas at Tyler poll that showed him beating all Democrats in Texas and leading Trump by 11 percentage points.

"Winning those 38 Texas electoral votes allows us not only to defeat Donald Trump in November of 2020, it forever changes the electoral landscape in the United States," O'Rourke told The Dallas Morning News. "This is incredibly important and we uniquely have the ability to follow through."

But the Emerson poll shows Biden ahead of O'Rourke by a 28% to 19% margin. Sanders was third with 16%, Massachusetts Sen. Elizabeth Warren had 14%, South Bend Mayor Pete Buttigieg was at 7% and California Sen. Kamala Harris had 5% support.

"Joe Biden has gone up 5 points since our April poll, and Beto O'Rourke has dropped a couple of points," Kimball said.

Dallas pastor Richie Butler, who was an aide to former Massachusetts senator and 2004 presidential nominee John Kerry, said Texans -- at the moment -- feel that the well-known Biden is the best choice to beat Trump, not O'Rourke.

"A lot of people feel that Beto has a better chance of getting across the finish line in a Senate race, not a race for president," Butler said.

Though O'Rourke was highly competitive in the poll, former Housing secretary and former San Antonio mayor Castro barely registered. At 2%, he trailed businessman Andrew Yang (3%) and was tied with New Jersey Sen. Cory Booker. All other candidates had 1% or less.

But about 41% of poll participants said they wanted Castro to drop out of the race and run for Senate, with 31% unsure and 29% saying he should stay put.

Democratic presidential candidate Julian Castro speaks during the 2020 Public Service Forum hosted by the American Federation of State, County and Municipal Employees (AFSCME) at UNLV on August 3, 2019 in Las Vegas, Nevada. (Ethan Miller / Getty Images)

Like O'Rourke, Castro is selling himself as a Democrat who could win Texas and slam the door on Trump.

"Not only can I win Texas in a general election, I'm the party's best hope to go out and get the 29 electoral votes in Florida and the 11 electoral votes in Arizona, in addition to recovering Michigan, Wisconsin and Pennsylvania," Castro told The News last week.

The Emerson poll, like most others, doesn't back up that boast.

Wide-open race against Cornyn

The Democratic race to challenge Cornyn for his Senate seat could be a bruising primary that's too early to handicap.

Hegar, the Round Rock businesswoman who has raised just over $1 million for her campaign, has failed to put distance between herself and rivals, both existing or potential.

West has not built out his organization and has yet to raise significant campaign dollars. Bell and Edwards are also in the early stages of their campaigns.

Hegar had the choice of running for Senate or going for a rematch against Rep. John Carter of Round Rock, who narrowly beat her last year.

Though Cornyn's approval rating is lackluster, analysts say he's in a good position for reelection.

"In and of itself his numbers show vulnerability," said veteran Republican consultant Bill Miller. "At present he doesn't have strong competition from the Democrats in the field. That could change, but right now he's in good shape."

If he announces a run for the Democratic spot against Republican U.S. Sen. John Cornyn (far right) on Monday, July 22, 2019, state Sen. Royce West (top left smaller mugshot), D-Dallas, would join (continuing clockwise) former U.S. Rep. Chris Bell of Houston, Houston City Council member Amanda Edwards and former Army helicopter pilot MJ Hegar of Round Rock in the field of those trying to unseat the longtime incumbent. (File/wires)

The issues

When asked what the most important issue is in casting their vote for president, there's a stark contrast among Texans.

The top three responses were immigration (30%), the economy (20%) and health care (14%), followed by social issues (9%), the environment (8%) and gun control (6%). The poll was conducted before the mass shootings in El Paso and Dayton, Ohio, that occurred within 24 hours of each other.

Republicans were heavily focused on immigration (48%) and the economy (26%) over other issues.

Democrats were split among health care (21%), immigration (15%), impeaching Trump (15%) and the economy (14%).

CORRECTION, 5:11 p.m., Aug. 6, 2019: An earlier version of this story had two charts that stated the wrong number of respondents and margin of error for the poll of Democratic voters. Emerson College polled 400 registered voters who said they would vote in the Democratic primary with a margin of error of plus or minus 4.9 percentage points.