ARGYLE, Texas — His famous name shadows George P. Bush, the only member of the dynastic political clan now in public office, as he enters the final days of an uphill campaign to unseat Texas’ attorney general.

To some Texans, the Bush family name is a badge of integrity, harking back to a bygone era of rectitude and respectful political debate. To others, it is the disqualifying mark of a Republican old guard that failed the party and betrayed its last president, Donald J. Trump.

Mr. Bush would like to make the campaign about the two-term Republican incumbent, Ken Paxton, whose serious legal troubles — including an indictment on securities fraud charges and a continuing federal corruption investigation — prompted high-profile Republicans to take him on in the primary. Mr. Bush made it to a runoff with Mr. Paxton that takes place on Tuesday.

A few years ago, Mr. Bush, whose mother is from Mexico and whose father was the governor of Florida, might have won the race handily, his aides believe, and then been held up as a prominent example of a new, more diverse generation of Republicans.

But that was before the ground shifted and his family spoke out publicly against Mr. Trump, in an unsuccessful effort to derail his bid for the presidency.

Mr. Bush broke with his father (Jeb), his uncle (George W.) and his grandfather (George H.W.) and aligned himself with Mr. Trump and his followers. The effort to distance himself from his relatives was captured in a campaign beer koozie that his campaign handed out last year, quoting Mr. Trump: “This is the Bush that got it right. I like him,” it says, beneath a line drawing of Mr. Trump shaking Mr. Bush’s hand.

The effort did not pay off. Mr. Trump endorsed Mr. Paxton, who had filed lawsuits seeking to overturn the 2020 election and had appeared with Mr. Trump at his rally in Washington on Jan. 6, 2021, before members of the crowd stormed the Capitol.

Some Texans say the political obituary has already been written for the Bush family, and see Mr. Bush, who is currently the state land commissioner, as its last flickering ember, with little of his forebears’ appeal.

“Daddy Bush was wonderful, wonderful, wonderful,” Carolyn Lightfoot, a member of the Daughters of the Republic of Texas, said of Mr. Bush’s grandfather. But the organization has criticized George P. Bush’s moves as land commissioner over his handling of the Alamo in San Antonio. Ms. Lightfoot said the Bush family and the party establishment were “trying to stuff him down our throats because of his Latino heritage.”

For all that the family’s importance may have faded among Texas Republicans, Mr. Bush may still emerge victorious in the runoff. A poll this month had Mr. Paxton’s support at less than 50 percent, and Mr. Bush trailing him by only a few percentage points. Donors have pumped new money into Mr. Bush’s campaign in the final stretch, hoping to push him over the top.

Mr. Bush has tried to refine and target his attacks on Mr. Paxton in recent weeks, after his campaign’s internal polling suggested that earlier efforts were hurting his own standing along with Mr. Paxton’s. And Mr. Bush has proudly invoked his family, both in a closing-message political ad and while speaking to audiences that might be unimpressed with the Bush name.

“It’s all about ethics,” Mr. Bush told a gathering of Republican women this month in Argyle, a town in the rapidly growing, largely Republican suburbs of Fort Worth. “When people say the last thing we need is another Bush, my response is, this is precisely the time that we need a Bush.”

As he barnstorms the state, Mr. Bush, 46, is invariably asked about his relatives, told about some fond memory of them, or challenged to reiterate his loyalty to Mr. Trump.

After the event in Argyle, a man in a cowboy hat waited outside for Mr. Bush to emerge so he could confront the candidate.

“Would you support for president the Republican nominee, even if it is Trump in 2024?” the man asked.

“Yeah, no, I would support him again,” Mr. Bush replied as he walked to his car, wearing black cowboy boots emblazoned with a White House seal and a reference to his uncle’s ranch in Crawford, Texas. “But we’ll see who comes out.”

At a Republican club event in Houston, held down the road from an apartment George W. Bush used to occupy in an area George H.W. Bush used to represent in Congress, George P. Bush delivered a speech attacking Democrats and Mr. Paxton. He promised to strengthen the state’s border with Mexico and to address Houston’s rising murder rate. He opened the floor to questions, but got a comment to start.

“I enjoyed watching you talk, because to me, you have all the mannerisms of Governor Bush,” a man told him, to laughter in the room. “Your hands are just like ‘Saturday Night Live.’”

Another attendee also made reference to his family. “I’ve heard people say that they’re not going to vote for you because they’re tired of the Bush dynasty,” said Doug Smith, a club member, echoing the views of some in the room. “How do you respond to those people?”

“I’ll never run away from being a Bush; I love my family,” he said. Most of the crowd applauded.

To live in Texas is to be exposed to the ubiquity of the Bushes, whose family name is borne by airports, roads and schools from Houston to Dallas to Midland. Both Bush presidents have their presidential libraries in the state. In Houston, there are even dog parks named for the canine companions of George P. Bush’s grandmother Barbara Bush, who died in 2018.

Exposed to a national spotlight from a young age, Mr. Bush has been hearing about his bright political future for decades. “The Republican convention is doubling as a dress rehearsal for a man Republicans talk about as an up-and-coming heir to the Bush legacy,” The Baltimore Sun wrote of him in 2000, referring to him as a “hunk” who could put “the passion in compassionate conservatism.”

But that is not the message Republicans want to hear now, Texas political consultants, donors and observers said.

“Everything was lining up to give him the brass ring, but the party changed too much,” said Brandon Rottinghaus, a professor of political science at the University of Houston. “The Republican base changed in such a fast way that many were left without a chair when the music stopped. Bush is a great example of that.”

Jay Zeidman, a longtime friend of Mr. Bush’s, said he believed that those shifts masked a dissatisfaction with the direction the party had taken. “There’s a lack of political courage in this state right now because of Donald Trump,” he said. “I think Americans and Texans are thirsty for some reversion back to what politics used to be.”

As he campaigns, Mr. Bush, who grew up in Florida, underscores his ties to Texas: Born in Houston, college at Rice University, a law career in the state. In an interview, Mr. Bush said he understood the legacy of his family as something Texan, as well as “quintessentially American and patriotic.”

“My role is to close the wounds of the past,” Mr. Bush said. “What I focus on are areas that I can control, and not focus on the areas that I can’t control. Because that would be futile.”

Mr. Bush has staked out hard-line positions that appeal to Republican primary voters on issues like the teaching of race and gender in schools. On immigration, he has urged Texas to formally invoke passages in the U.S. Constitution referring to “invasion,” a step toward the state seizing war powers and a move that Mr. Paxton and Gov. Greg Abbott have so far avoided making. He has said there was “fraud and irregularity” in the 2020 election, though he did not believe it changed the outcome.

He has challenged Mr. Paxton to debate him on issues, but the two have not shared a stage during the campaign. Mr. Bush contrasts his willingness to field questions from reporters and from a variety of audiences with Mr. Paxton’s practice of rarely holding news conferences or taking challenging questions.

Mr. Paxton’s campaign declined a request for an interview.

“Texas voters have made it clear that they are sick and tired of the Bush family dynasty and their RINO establishment donors playing kingmaker in Texas politics,” said Kimi Hubbard, a Paxton campaign spokeswoman, using an acronym meaning “Republican in name only.”

Mr. Bush was careful in an interview with The New York Times not to question the shifts in the Republican Party that have made his run for office more difficult. He said the concerns of party voters were largely the same as when he first ran for land commissioner in the 2014 election: “Concerns on my family, concerns on crime, border security.”

Have voters’ feelings about the Bush dynasty hurt him? “I wouldn’t say so,” he said. “I’ve won.”

A significant number of Republicans polled in Texas say they would not support Mr. Bush because of his family background. But his lineage is not simply a liability.

In this month’s poll by The Dallas Morning News and the University of Texas at Tyler, people planning to vote in the primary runoff for attorney general were asked what they liked about their chosen candidate. One of the top factors Mr. Paxton’s supporters mentioned was that he was not a Bush. But about the same share of Mr. Bush’s backers said they were drawn to him specifically because he was a Bush.

Mr. Bush has drawn financial support from his family’s network, including six-figure checks from some longtime Bush supporters and more than $100,000 directly from his uncle George W. Bush, campaign finance records show.

A week before the runoff, outside an early voting location in his grandfather’s old congressional district in Houston, Mr. Bush’s family name loomed large for Republican voters, both for and against.

“We support George P.,” said Julie Treadwell, 50, who had just voted with her 18-year-old daughter. “We want to get back to that,” she said of his family and what they represented to her: “Conservative Republicans that are more even-keel and levelheaded.”

Darla Ryden, 59, who overheard Ms. Treadwell’s remarks, waited until she had walked away to her car before describing her own views, which she said were just the opposite.

“I was all for George Bush, daddy and son, but now I feel, with the Bushes, it’s more about power than it is about people,” Ms. Ryden said. She voted for Mr. Paxton in the runoff and supported him in the first round of the primary as well, she said, despite “his own struggles.”

“The Bushes?” she added. “It’s done.”