Texas state Rep. James Talarico, the Democratic nominee for U.S. Senate in the Lone Star State, conceded Wednesday that some past statements during debates over transgender policy "missed the mark," but accused his Republican opponent Ken Paxton of "intentionally clipping my cringey comments to distract from his career of corruption."

Republicans are "worried about the movement that we're building here in Texas," Talarico said, hours after Republicans overwhelmingly chose Paxton, the state attorney general, as their Senate nominee. 

"I think national Republicans see the momentum behind this movement, and they're worried we're going to win Texas in November," Talarico told CBS News' Ed O'Keefe in his first network news interview since Tuesday's runoff. 

He told O'Keefe that his campaign has been attracting huge crowds across the state, and he believes his support is expanding beyond the Democratic voter base.

"We've had tens of thousands of Texans showing up to rally with us. And I can't tell you the number of people who've come up to me at the end of these events and whisper, 'I'm not a Democrat,' like they're in the witness protection program," Talarico said. "I think we're reaching people who don't normally vote for Democratic candidates. We're also reaching a lot of first-time voters, a number of young people, who've been showing up at our events, who've been signing up to volunteer and donate."

Paxton won the Senate primary runoff for the GOP nomination on Tuesday night in a landslide, defeating Sen. John Cornyn, the incumbent senator who has represented Texas since 2002. 

Senate GOP leadership had thrown its support behind Cornyn — a prolific fundraiser who has brought in more than $400 million for Republican Senate candidates — and viewed him as a better candidate to take on Talarico in the general election. Now, with Paxton as the nominee, Republicans will have to dump large sums of money into the race to support his lackluster fundraising and combat an onslaught of attacks from the left over a number of ethics scandals. 

For his part, Talarico has also flexed his fundraising chops, bringing in $27 million in the first quarter. He defeated Rep. Jasmine Crockett in March for the Democratic nomination. 

Talarico said "Texans are drowning."

"We can't afford the basics, we can't afford groceries or gas or insurance or housing or childcare or prescription drugs, and Ken Paxton clearly has no solutions to offer us," he said. "So, while he divides us with these same tired culture war fights, I'm going to keep bringing Texans together to take on his corruption and lower our costs."

To drive down costs in Texas, Talarico proposes ending tariffs, suspending the federal gas and diesel taxes and capping prescription drug prices. He said along with the cost of housing, childcare and insurance, "all of these things are conspiring to hold Texans back." 

Talarico echoed a message he's shared since the beginning of the campaign — that the fight isn't left versus right, it's "top versus bottom."

"We're all getting screwed. None of us can afford anything. None of us can get ahead, no matter how hard we work, and that's because the system, this political system and economic system, are rigged against us," he said. "It's been rigged for 50 years by billionaire megadonors and their puppet politicians — like Ken Paxton."

The Democratic nominee argued that corruption is the "central problem in our country right now."

"I think we're living in an era of corruption, and I don't just mean illegal activity, although we're certainly going to hear a lot about that, given that Ken Paxton is the nominee. But I'm talking about corruption in the deeper sense, the rotting of our systems from the inside out, the decaying of long-held values and institutions, and we see it everywhere," he said. 

Talarico said "this campaign is really about taking on that corruption head-on."

The general election is expected to be among the most expensive Senate races in the country. The race could also determine control of the Senate, as Democrats look to flip the chamber. Republicans currently have 53 seats in the Senate, meaning Democrats would need to net four to retake the majority. And despite a difficult map, Democrats see Paxton's nomination as bringing them one step closer to a long-sought victory in Texas. No Democrat has prevailed in a Senate race in Texas since 1988, when Sen. Lloyd Bentsen won reelection. The Cook Political Report shifted its rating from Likely Republican to Lean Republican after Paxton's victory. 

Talarico hopes to defeat Paxton by adding Cornyn supporters to his base of Democratic voters. He appealed to them Wednesday, thanking Cornyn for his "many years of service to our state" and noting that while they don't agree on every issue, "we both still believe in public service." 

"We both still believe that these positions of public trust should be about serving our neighbors, not serving ourselves," Talarico said. "And so to any of Senator Cornyn's supporters who still believe in that old-fashioned idea of service, I want them to know they have a place in our campaign."

As the general election fight gets underway, Talarico has faced intense criticism from Paxton and other Republicans, like GOP Sen. Ted Cruz of Texas, who called Talarico "absolutely extreme" on his podcast overnight, adding, "What makes Talarico dangerous — and that worries me — is he packages it in this very nice 'aw, shucks' demeanor, and he sounds like a preacher."

Talarico, a Presbyterian seminarian who calls himself a progressive Christian, has faced attacks from the right over previous comments, including on veganism and gender. 

Pressed on a comment he made that God is nonbinary, Talarico said he was being "intentionally provocative," but added that "what it means is that God can't be defined by human categories." On his comments about gender, Talarico said "I know there are two sexes, men and women. I also know there's a very small percentage of people who have these chromosomal abnormalities, and I believe they deserve to be treated with dignity and respect."

Asked by O'Keefe about the criticism surrounding his diet, Talarico said he is not a vegan, noting that he's an eighth-generation Texan who's "been eating barbecue since before Ken Paxton's first indictment."

"If all they have is lying about me being a vegan, I feel pretty good about our chances this November," he said.