SAN ANTONIO — The race for Texas attorney general is asking Republicans to determine how many indictments and allegations of corruption are too many. The answer may be there is no limit — so long as the candidate has an endorsement from former President Donald J. Trump.

Ken Paxton, the Trump-backed attorney general, was indicted and arrested on criminal securities-fraud charges that are still pending. He has faced calls for his resignation after several of his top aides claimed he abused his office by helping a wealthy donor. And he has been serving as the state’s top lawyer while under threat of a possible new indictment, as the F.B.I. investigates the abuse-of-office and bribery accusations.

“The voters of Texas will tolerate a great deal,” said State Senator Kel Seliger, a moderate Republican who is a former mayor of Amarillo. “They think if somebody is ideologically in sync with them, that’s what matters. I would have thought in Texas that moral example is more important, but apparently it’s not.”

In the pre-Trump era, indictments and investigations by federal law enforcement could have been fatal to a Republican campaign. But Mr. Trump has instilled a deep mistrust in government institutions like the F.B.I. Mr. Paxton took the unusual step of authorizing an investigation of an F.B.I. investigation — he appointed a special prosecutor to look into the federal probe of the wealthy donor, an Austin real estate investor named Nate Paul whose home and offices were raided by federal agents.

The litany of allegations against Mr. Trump has allowed acolytes like Mr. Paxton to claim that they, too, are victims of a government conspiracy.

“That’s the Biden F.B.I., the Biden D.O.J.,” Mr. Paxton said in a recent interview with a Fox News reporter. “They were under investigation by my office. I don’t know what they are going to do. All I can tell you is that we were doing the right thing. We are going to continue to do the right thing. I don’t control what the Biden White House does.”

Since the 2020 election, Mr. Paxton has made himself among the nation’s foremost Trump defenders, filing an audacious lawsuit with the Supreme Court seeking to delay certification of the results in Georgia, Michigan, Pennsylvania and Wisconsin. He spoke at the Jan. 6 rally in Washington that preceded the attack on the United States Capitol, won an endorsement from Mr. Trump and earned praise from him at the former president’s rally outside Houston. And he has overlooked the fact that, although he has claimed otherwise, the federal abuse-of-power investigation began under Mr. Trump’s F.B.I., not Biden’s.

In the Republican primary on Tuesday in Texas, Mr. Paxton won 43 percent of the vote, a soft showing for an incumbent but one indicative of the three well-funded challengers who saw him as politically vulnerable. Since Texas requires primary candidates to win a majority of the vote to advance to the general election, Mr. Paxton faces a May 24 runoff against the scion of the most famous family in modern Texas politics: George P. Bush, the state’s land commissioner who is the nephew of one president and the grandson of another.

The Paxton-Bush runoff crystallized immediately as a contest between an incumbent with ethical and legal issues and a challenger who cannot escape the establishment brand of his family name. In Texas Republican circles, some operatives cast the race as prison stripes versus pinstripes.

Mr. Paxton has withstood his legal woes by delivering on the issues that drive Texas conservatives. He’s used his office as the state’s chief culture-war litigator — defending the new Texas abortion law, suing the Biden administration to force the federal government to continue building the border wall and joining a right-wing push to criminalize medical care for transgender youth. Days before the primary, he issued an opinion stating that certain medical treatments for transgender children could be considered child abuse, treatments that doctors describe as gender-affirming care.

Mr. Paxton did not take long to attack Mr. Bush as a symbol of the moderate conservative politics that Mr. Trump has all but excised from the Republican Party.

“What has happened with performance by the Bushes over the last decade, it’s been disappointing,” Mr. Paxton said Wednesday during an interview on a conservative talk radio show in Lubbock. “I think a lot of Republicans have had enough of it. The Bushes have had their chance. It’s time for the dynasty to end.”

That a top elected official in Texas could make such a stunning anti-Bush remark and face no political consequences illustrates just how loyal Texas Republicans have become to Mr. Trump and Trumpism.

In Mr. Bush, Mr. Paxton has a near-perfect foil for a runoff election that is likely to have half or less the turnout from Tuesday’s primary. Mr. Bush has been an enthusiastic supporter of Mr. Trump, but his father, Jeb Bush, the former governor of Florida and a 2016 presidential candidate, and his uncle, former President George W. Bush, have been harsh critics.

At a debate last month, the younger Mr. Bush said President Biden was the rightful winner of the 2020 election and called Mr. Paxton’s lawsuit to block the election results “frivolous” — statements Mr. Paxton’s campaign is using to attack Mr. Bush as insufficiently conservative.

Mr. Bush said in radio interviews in recent days that he has contacted Mr. Trump’s advisers to suggest that he switch his endorsement from Mr. Paxton. A Trump aide said that was extremely unlikely. And Mr. Paxton said he spoke with Mr. Trump himself and extracted a pledge that the former president would continue to support him through the runoff.

Still, Mr. Bush, whose father was savaged as “low energy” by Mr. Trump during the 2016 presidential campaign, is not conceding Mr. Trump’s support. Last summer he distributed red koozies with a silhouette of himself shaking hands with Mr. Trump and a quote from the former president: “This is the only Bush that likes me! This is the Bush that got it right.”

Mr. Bush signaled he will lean into Mr. Paxton’s ethical and legal issues, which have long been talked about in Texas political circles. In 2014, after Mr. Paxton was first elected attorney general but had not yet taken office, he was accused of taking a $1,000 pen that belonged to another lawyer. (He later returned it and said the episode was a simple mistake.) The State Bar of Texas is also investigating whether Mr. Paxton committed professional misconduct by challenging the 2020 presidential election results.

In his own radio interview in Lubbock, Mr. Bush said the F.B.I. investigation and the securities-fraud case “are a matter of public record and should be discussed.”

Mr. Bush’s campaign spokeswoman did not return repeated messages this week. Mr. Paxton declined to be interviewed.

Mr. Paxton has denied wrongdoing in the securities case and has rejected claims that he accepted bribes in office. Last August, his office produced a 374-page report that cleared him of any wrongdoing and said there was “no evidence” he had accepted a bribe. “A.G. Paxton committed no crime,” the report issued by his office stated.

There have been signals that Mr. Paxton’s litany of controversies has tested the limits of Texas Republicans’ patience with him. Representative Chip Roy, a conservative who used to work for Mr. Paxton, called for his resignation in 2020. Along with Mr. Bush, Mr. Paxton’s primary challengers included Representative Louie Gohmert, who gave up a safe East Texas congressional seat to run against him, and Eva Guzman, who resigned from the Texas Supreme Court to challenge him in the primary.

During his campaign, Mr. Gohmert predicted Mr. Paxton would face a new federal indictment after winning the Republican nomination and lose the general election to a Democrat. If Mr. Paxton indeed wins the nomination but is defeated in November, it would be a devastating first for Republicans: No Democrat has won any statewide office in Texas since 1994.

In the Democratic primary for attorney general, Rochelle Garza, a South Texas civil rights lawyer, garnered the most votes and is headed for a runoff. Her Democratic opponent remained unclear. The third-place vote-getter, Lee Merritt, a civil rights lawyer, said in a statement he was not ready to concede to the second-place candidate, Joe Jaworski, a former mayor of Galveston, because military and other ballots were still being counted. Ms. Garza said she was confident the attorney general’s office could be flipped from red to blue. In 2018, Mr. Paxton won re-election by narrowly defeating his opponent, Justin Nelson, by 3.56 percentage points.

Mr. Paxton has brushed off any suggestion of a Democratic victory in the fall. “In this country, allegations don’t convict you,” he said in the Lubbock radio interview.

Mr. Paxton’s aides said Texas Republicans don’t care about the allegations and controversies surrounding his office. They claimed credit for attacking Mr. Gohmert and Ms. Guzman in order to allow Mr. Bush to advance to the runoff. After the Paxton campaign attacked Ms. Guzman in television advertisements in the closing days before the primary, she dropped from winning 21 percent of the vote during the early-voting period to just 14 percent of the vote on Tuesday.

“These ads clearly cost Eva a spot in the runoff,” Dick Weekley, the senior chairman of the mainstream Republican group Texans for Lawsuit Reform, which endorsed Ms. Guzman, wrote in an email to supporters after the primary.

Both Mr. Paxton and Mr. Bush are certain to continue to pitch themselves as the true steward for Trump supporters among Texas Republicans.

“It’s easy for me to say that I wouldn’t grovel for the Trump endorsement,” said Jerry Patterson, Mr. Bush’s predecessor as land commissioner and a Republican who is anti-Trump but is backing Mr. Paxton. “It’s just damn distasteful for George P. At some point you just have to have some pride in your own integrity.”

Yet Mr. Patterson said he has no problem with Mr. Paxton doing Mr. Trump’s bidding about the 2020 election and constantly stressing his Trump bona fides.

“For Paxton that came naturally,” Mr. Patterson said. “It’s not contrived.”

Kirsten Noyes and Kitty Bennett contributed research.