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Media outlets say both candidate have distorted facts in their attack ads.

For his part, Mr. Dewhurst dismissed the University of Texas Internet survey of 800 Texas voters as “nonscientific” and pointed to other recent independent surveys, including one conducted in mid-April by Public Policy Polling that showed him with a 15 percentage point lead.

“I always laugh when Mr. Cruz talks about being a fighter,” Mr. Dewhurst said. “The boy doesn’t have the slightest ideas of what fighting is about.”

Mr. Dewhurst has touted his experience as Air Force officer, a CIA agent and a former businessman, as well as a statewide political career that began in the 1990s as the commissioner of the Texas General Land Office.

“The more money and more bravado, the weaker people are,” he said in an indirect jab at Mr. Cruz. “It’s far less important what [candidates] say than what they’ve done and will do.”

Running for statewide office in Texas, which includes about 20 media markets, isn’t cheap, and with at least $5 million in outside money already spent on the race, the primary could be one of the most expensive in history.

Texas Conservatives Fund, a so-called super PAC, has spent more than $1.7 million in media ads opposing Mr. Cruz, says, a nonpartisan website that tracks money in politics.

Meanwhile, a super PAC affiliated with the anti-tax Club for Growth says it has spent almost $2 million in the past couple of weeks on efforts to oppose the Mr. Dewhurst’s candidacy. The group has endorsed Mr. Cruz in the race.

Mr. Cruz has called the Texas primary “a battle for the heart and soul of the Republican Party.”

“I think the tea party movement is the most exciting thing to happen in politics in decades,” said Mr. Cruz on Monday evening. “The focus is on taking over the Republican Party.”

But Mr. Dewhurst calls Mr. Cruz’s characterization of the race “baloney.”

“You don’t have moderates in Texas. We’re all conservatives,” he said.