Washington (CNN) Republicans outraged at Russia's alleged meddling in the US election -- and frustrated at some in their own party for not taking on the issue more aggressively -- are turning to an unlikely new source of pressure for action: Democrats.

In a letter to Senate Majority Leader Mitch McConnell Sunday, four senior GOP and Democratic senators appealed for a select committee to investigate CIA findings that Russia hacked the elections to help Donald Trump. The idea has so far been rejected by McConnell and House Speaker Paul Ryan, who instead want to see current committees look into the matter.

One of the signatories of the letter, Arizona Republican John McCain, also appeared on Sunday news shows to press his case with dire warnings, while Democratic victims of the hacks gave some of the first personal descriptions of their experience.

McCain put the blame on Moscow for damage to the integrity of the vote, saying that Russian election-related hacks threaten to "destroy democracy" and faulted the American response as "totally paralyzed."

"This is serious business. If they're able to harm the electoral process, they may destroy democracy, which is based on free and fair elections," he told CNN's Jake Tapper on "State of the Union." "This is the sign of a possible unraveling of the world order that was established after World War II, which has made one of the most peaceful periods in the history of the world."

Trump has continued to reject the conclusion of the Intelligence Community that Russia hacked Democratic institutions over the summer, as well as more recent CIA assessments that the purloined emails were shared in order to hurt Democratic nominee Hillary Clinton's chances.

Trump's team has expressed concern that accepting the CIA conclusion would rattle the legitimacy of his win, even as the Electoral College is set to endorse it Monday. Trump's incoming chief of staff, Republican National Committee chairman Reince Priebus, on "Fox News Sunday" said that "the reality of all of this and all of these players that are spinning these reports are doing it for a political purpose, which is to delegitimize the outcome of the election."

The Republican leaders of both chambers of Congress have seemed interested in denying the issue the more prominent platform that creating a select committee to investigate the Russian breaches would provide.

But in the letter to McConnell, McCain, along with fellow GOP Sen. Lindsey Graham, incoming Senate Democratic Leader Chuck Schumer and Sen. Jack Reed, the top Democrat on the Senate Armed Services Committee, wrote that, "Cyber is the rare kind of all-encompassing challenge for which the Congress's jurisdictional boundaries are an impediment to sufficient oversight and legislative action."

They continued, "Only a select committee that is time-limited, cross-jurisdictional, and purpose-driven can address the challenge of cyber."

McCain's harsh words were complemented by the account of two senior Democratic operatives whose emails were leaked after the hacks.

Donna Brazile, the interim chair of the Democratic National Committee, where hackers stole staffers' emails, said it was "a very intrusive process."

"The emails were weaponized," Brazile said on ABC's "This Week." "The release of stolen, hacked emails caused a lot of confusion and of course it disrupted our daily campaign life."

She said the FBI warned the DNC of the Russian hacks in May.

"But in terms of helping us to fight -- we were fighting a foreign adversary in the cyber space. The Democratic National Committee, we were no match," Brazile said.

She said the hacking attempts didn't end after the first batch of DNC emails were obtained. Hackers kept attempting to breach the DNC all the way through the November 8 election, she said.

"They came after us daily. Hourly," Brazile said.

Brazile also complained that the federal government didn't do more to help the DNC fight off foreign attackers.

"We never felt comfortable. We didn't know what was coming next," she said. "And, you know, this is not just about computers; this is harassment of individuals, it's harassment of our candidates, harassment of our donors. We had stolen information, personal information. People were personally harassed."

John Podesta, the Clinton campaign chairman who watched the stolen contents of his Gmail account slowly posted on WikiLeaks through October, said the only time he was contacted by the FBI was two days after WikiLeaks began posting his emails.

"That was the first and last time I talked to the FBI," Podesta said Sunday on NBC's "Meet the Press."

However, federal investigators tried to warn the Democratic National Committee about a potential intrusion in their computer network months before the party moved to try to fix the problem, US officials briefed on the probe previously told CNN. The DNC brought in consultants from the private security firm CrowdStrike in April. And by the time suspected Russian hackers were kicked out of the DNC network in June, the hackers had been inside for about a year.

Podesta also noted on "Meet the Press" that his emails were posted beginning two days after the "Access Hollywood" tape in which Trump bragged about sexually assaulting women became public.

"Those things might not have been a coincidence," Podesta said sarcastically.

"Two days later, the FBI contacted me, and the first thing the agent said to me was, 'I don't know if you're aware, but your email account might have been hacked,' " he said.

Still, there isn't complete bipartisan harmony between those who would like a more comprehensive response to Russia.

McCain mocked President Barack Obama's statement Friday that he had personally told Russian President Vladimir Putin this fall to "cut it out" once the hacking allegations came to light and that this had curbed the activity.

"There's no doubt they were interfering and no doubt it was a cyber-attack," McCain said. "The question now is, how much and what damage and what should the United States of America do? And so far, we've been totally paralyzed," he said.

"I'm sure that when Vladimir Putin was told quote 'cut it out' unquote, I'm sure that Vladimir Putin immediately stopped all cyber activities," he deadpanned.

McCain continued, "The truth is, they are hacking every single day."

Most lawmakers -- including congressional leaders -- have accepted the intelligence community's assessment that Russia is responsible for the hacks and the hacks were intended to help Trump win the election.

But Trump and his top aides, angry at what they see as an attempt to undermine the legitimacy of his victory, have cast doubt on those assessments.

CIA Director John Brennan said in an unclassified memo to CIA staffers on Friday that FBI Director James Comey and Director of National Intelligence James Clapper now both concur with his agency's finding.

Priebus on Sunday called on Brennan, Comey and Clapper to publicly address their findings.

"It sure would be nice to hear from everybody. I mean, if there is this conclusive opinion among all of these intelligence agencies, then they should issue a report or they should stand in front of a camera and make the case," Priebus said.