Republicans in Georgia have defied Donald Trump to vote overwhelmingly for two of the party officials whom the former president holds most accountable for his 2020 presidential election defeat.

Brian Kemp will seek a second term as Georgia governor after a convincing win over his main rival David Perdue, while Brad Raffensperger, Georgia’s secretary of state, will remain the Republican candidate after defeating a challenge from Jody Hice.

Trump had opposed both Kemp and Raffensperger, whom he blamed for certifying Joe Biden’s 2020 election victory in the state. During the primary campaign, Trump called Kemp a “turncoat” and a “coward”.

The results marked the most bruising defeats yet for the former president of this primary season, during which the candidates he has backed have achieved mixed results.

Some Trump-endorsed candidates have chalked up notable victories, such as JD Vance in Ohio and Doug Mastriano in Pennsylvania.

But others have fallen short, such as Madison Cawthorn in North Carolina. Mehmet Oz, the daytime television star who Trump backed for senator in Pennsylvania, is heading for a recount in his race against David McCormick, the former hedge fund boss.

Kemp, who will face Stacey Abrams in November’s midterm election, spent Monday campaigning alongside Mike Pence, the former vice-president, to whom Trump has also attached blame for his 2020 election loss.

“I’ve had a great relationship with Mike Pence; I had a great relationship with President Trump,” Kemp said on Monday. “I’ve never said anything bad about him, I don’t plan on doing that.

“I’m not mad at him, I think he’s just mad at me. That’s something I can’t control.”

Perdue conceded on Tuesday evening, saying: “Everything I said about Brian Kemp was true, but here’s the other thing I said was true — he is a much better choice than Stacey Abrams, and so we are going to get behind our governor.”

A former US senator, Perdue had spent the last stage of the campaign engulfed in a dispute over remarks he made on Monday about Abrams.

“[She] said Georgia’s the worst place in the country to live,” he told supporters of Abrams, who is black and has lived in Georgia since she was a teenager. Perdue claimed Abrams was “demeaning her race” with comments she made in 2018 that people should not have to go into agriculture or hospitality to earn a living.

Trump’s failure in Georgia was amplified by Raffensperger’s win. When he claimed victory late on Tuesday night, Raffensperger did not mention Trump by name but made reference to the pressure the former president had applied to get him to change the 2020 result. “Not buckling under the pressure is what the people want,” he said.

Some of Trump’s other favoured candidates prevailed, however, including Herschel Walker, a former American footballer and Trump ally, who easily clinched the Republican nomination for a US Senate seat from Georgia. He will face the incumbent, Democratic candidate Raphael Warnock, in the general election this autumn.

In Arkansas, Sarah Huckabee Sanders, a former Trump White House press secretary, handily won the Republican primary for governor, an office also previously held by her father, the two-time presidential hopeful Mike Huckabee.

Trump’s backing also held sway in Texas, where Ken Paxton beat George P Bush, one of the youngest members of the US political dynasty, in the race to become the Republican nominee for state attorney-general.

Paxton, who supported Trump’s false claims to have won the election, secured his party’s backing despite being indicted for securities fraud seven years ago, as well as more recent accusations of bribery while in office. He has denied all of the allegations.

Texas Democrats were set for a longer wait to find out who their candidate for the state’s 28th congressional district would be. Henry Cuellar, the incumbent and the only anti-abortion Democrat in the House of Representatives, has faced a challenge from leftwing activist Jessica Cisneros.

After several hours of voting, Cisneros held the lead in their run-off, but with many Cuellar-supporting areas still to report most of their votes.