“We delivered the largest tax cuts in American history.”

But Texas is not just the incidental backdrop for an event aimed at a national television audience; it took center stage for much of the convention. The focus was a sign of the growing concern among Republicans that a wave of Democratic enthusiasm could even breach Texas’s red wall in November.

Much of the attention was on Ken Paxton, the state’s attorney general and nominee for U.S. Senate, who was trailing James Talarico, his Democratic opponent, in many polls. President Trump vowed in his speech on Wednesday to come back to Texas “twice or three times” to support Mr. Paxton, while also acknowledging the candidate did not always impress.

“He may not dress right,” the president said. “He may not talk exactly perfectly, and he may not be the best-looking guy I’ve ever seen.”

There were also speeches from three congressional candidates who are running in districts specifically designed by the Texas legislature last year to favor a Republican. Each is facing a tough election fight despite the new lines.

One of them, Tano Tijerina, who is trying to oust Representative Henry Cuellar, a Democrat in South Texas, delivered a rousing, red-meat speech, his eyes shadowed by a large cowboy hat. Loud cheers erupted as he described himself as “beef-eating, truck-driving, gunslinging,” prompting him to stop midsentence and point to the sky.

There were backdrops of West Texas on the screen and more cowboy hats than usual in the arena, which usually hosts the Dallas Mavericks of the N.B.A. and the Dallas Stars of the N.H.L. Of the speeches reserved for “everyday Americans,” several came from Texans who spoke of their experience on the Mexican border or resisting gender-transition care in Texas hospitals.

Lt. Gov. Dan Patrick, who is running for a fourth term, described Texas as the first bulwark against a liberal invasion of communists and socialists, a frequent refrain among speakers throughout the convention. A loss in Texas, he warned, would cascade across the country.

“In Texas, we have a saying: We have to get it right because there’s not another Texas to move to in November,” Mr. Patrick said on Thursday. “We have to get it right because there’s not another America to move to.”

And Senator Ted Cruz, who is not on the ballot in November, put the spotlight on an issue that has been a focus of Texas Republicans this election season: Islam.

State leaders have taken steps over the last year to curtail the activities of Muslim organizations. In a dark speech on Thursday that assailed Muslim Democrats and reflected a growing tolerance for Islamophobia among some party officials, Mr. Cruz falsely portrayed such officials as seeking to impose strict Islamic religious rules.

“Shariah law has no place in America,” Mr. Cruz said, drawing huge applause from the audience on Thursday. “We need to stand together and stop it now.”

But Newt Gingrich, the former Republican speaker of the U.S. House, said the party should not be too worried about losing in Texas, even in Mr. Paxton’s race with Mr. Talarico.

“We’ll be fine,” he said as he walked through the convention on Wednesday. “Texas has a cyclical pattern of the bright, articulate new hope, who raises an amazing amount of money. And then Texans realize, ‘Why, he’s a Democrat!’”