(CNN) Special counsel Robert Mueller found that no one in the Trump campaign conspired with the Russian government in 2016 -- but Democrats are not ready to accept that finding.

In interviews since Attorney General William Barr issued his four-page letter on Sunday , Democrats have refused to accept that determination, saying there's ample evidence of Trump campaign and Russia contacts that may not have risen to the level of criminal conduct. They are demanding the full release of the Mueller report to determine what else the special counsel found, and they say they will continue investigating ties between Trump and Russia.

But that strategy risks political backlash for Democrats if they are viewed as overreaching and probing into an area that has already been exhaustively investigated by a special counsel whose investigation turned up no criminal wrongdoing.

"What I accept was there was apparently no criminal conspiracy ... with the Russians," Rep. Gerry Connolly, a Virginia Democrat, told CNN. "That doesn't mean there wasn't a lot of activity with the Russians that ranges from unsavory to treacherous."

Republicans say Mueller's finding that there was no conspiracy with Russia is vindication for the President -- and a clear sign it's time for Democrats to move on. Trump's "no collusion" assertion now appears to have been confirmed by the special counsel, and it's a message that Trump and congressional Republicans are sure to use to bludgeon Democrats heading into the 2020 campaign season.

House Minority Leader Kevin McCarthy, a California Republican, and Minority Whip Steve Scalise, a Louisiana Republican, called for House Intelligence Chairman Adam Schiff's resignation from the committee on Monday -- payback after Schiff and the panel's other Democrats pushed former Chairman Devin Nunes, a California Republican, to recuse himself from the panel's Russia investigation in 2017.

"When you look at the claims that they've made, Chairman Schiff said he had more than circumstantial evidence that there was collusion. Whether he was misleading people or he was misled himself, he ought to be held accountable," Scalise told reporters. "A lot of people, I think, should be angry today that for two years they've had people misleading and lying to them, saying there was collusion when there wasn't."

House Democrats, who have already launched numerous investigations into the President, are now forced to reckon with how aggressively they want to pursue questions surrounding Russia, or put that aside and focus on other issues, like obstruction of justice, where Mueller wrote the evidence "didn't exonerate Trump" and left the decision to prosecute to the attorney general.

The House Intelligence Committee had been scheduled to have former Trump business associate Felix Sater testify publicly on Wednesday, but the panel announced Monday that it was postponing that appearance in order to keep the focus on congressional Democrats' efforts to obtain Mueller's full report and underlying evidence.

Schiff told reporters Monday that his panel's investigation into Trump -- which touches on Russia, money laundering and other issues surrounding Trump's finances -- would continue.

"Our investigation has always focused on counterintelligence issues, that is, is the President or anyone around him compromised in some way? That work has to go on," the California Democrat said. "We need to look at, for example, into the financial issues: Was the President driven during the campaign and to this day by financial interests consummating a lucrative real estate deal ... or any other illicit purpose? That work has to go on."

He also brushed aside the calls from Republicans for him to step down.

"I'm more than used to attacks from my GOP colleagues, and I would expect nothing less," Schiff said.

House Speaker Nancy Pelosi dismissed questions from reporters about whether she still has confidence in Schiff. "Oh, please," the California Democrat responded.

The House Judiciary Committee, which opened a sprawling investigation into Trump, his administration and his businesses earlier this month, was supposed to speak to Sater on Thursday behind closed doors. It wasn't immediately clear whether that interview would be postponed, too.

Democrats on the Judiciary panel held a conference call Sunday and huddled in the Capitol on Monday to discuss their next steps. House Judiciary Chairman Jerry Nadler, a New York Democrat, declined to comment as he entered Monday's meeting.

Later on Monday evening, the chairs of six House committees wrote a letter to Barr demanding the release of Mueller's full report by April 2.

"Your four-page summary of the Special Counsel's review is not sufficient for Congress, as a coequal branch of government, to perform (its) critical work," the letter stated. "The release of the full report and the underlying evidence and documents is urgently needed by our committees to perform their duties under the Constitution."

Chairmen facing criticism

Democrats' past statements on Trump and Russia are facing new scrutiny. The Trump campaign on Monday circulated a list of Democratic statements made over the past two years in which Democratic lawmakers claimed evidence of collusion, and the campaign has been sending similar fundraising pitches.

Committee chairmen probing Trump -- Schiff and Nadler in particular -- have been among the biggest targets for Republican criticism since Barr's summary was released.

Pelosi was among the Democrats questioning whether Trump was compromised by Russia, saying in a January statement following Roger Stone's indictment: "In the face of 37 indictments, the President's continued actions to undermine the Special Counsel investigation raise the questions: what does Putin have on the President, politically, personally or financially?"

Pelosi's office said Monday that she stands by statement, saying, "Yes; this is why we need to see the underlying documents."

Later on Monday night, Pelosi told a meeting of congressional leadership that the Democratic caucus should stop focusing on the Mueller probe and Russian interferences, two sources told CNN. Pelosi and her top congressional lieutenants view the matter as a distraction and believe they should focus on pocketbook issues, the sources said.

Schiff, Nadler and Oversight Chairman Elijah Cummings issued a joint statement Sunday that noted Mueller declined to prosecute on a conspiracy to join with Russia's "online disinformation and hacking and dissemination efforts," while arguing they still need to know more from Mueller's report and underlying evidence.

"Although we have confidence that Special Counsel Mueller made the right prosecutorial judgment in these two specific areas — notwithstanding the very public evidence of Trump campaign contact with and willingness to receive support from Russian agents — it will be vital for the country and the Congress to evaluate the full body of evidence collected by the Special Counsel, including all information gathered of a counterintelligence nature," the chairmen said.

Other Democrats have echoed that sentiment, noting that Mueller did find Russia made an effort to reach out to the Trump campaign and that numerous contacts between Trump associates and Russians have already been established, even if it stopped short of criminality.

"Look, it may well be the case — and we're guessing because we haven't seen the report, that the special counsel has concluded there's insufficient evidence to charge the President or members of his campaign with conspiring with the Russians," said Rep. David Cicilline, a Rhode Island Democrat on the House Judiciary Committee. "And at the same time, there is evidence that, in fact, conspiracy or collusion occurred at some level."

Democrats see 'outstanding questions'

Rep. Ro Khanna, a California Democrat on the House Oversight panel, said there are "still outstanding questions" on Trump-Russia connections, adding that it's unclear whether Mueller looked into Trump's actions over Russia during his presidency. Khanna said he accepts Mueller's findings, but also that a "prosecutor can have a significant amount of evidence and say that it just doesn't rise to a crime ... that doesn't mean there wasn't misconduct."

House Democrats' immediate goal is to get the full Mueller report public and obtain the underlying evidence the investigation collected from 500 search warrants, almost 300 warrants for electronic data and 13 requests to foreign governments that took place during the special counsel's 22-month investigation.

And while they are still questioning Mueller's findings on a Russian conspiracy, they are seizing on what Mueller's report says about obstruction of justice: "While this report does not conclude that the President committed a crime, it also does not exonerate him."

Barr wrote in his summary that Mueller kicked the decision to prosecute on obstruction to the leaders of the Justice Department, and he and Deputy Attorney General Rod Rosenstein determined they could not bring a case. Now Barr — who wrote before joining the Justice Department that the obstruction case was "fatally misconceived" — will be forced to defend that decision on Capitol Hill, where Nadler said he expects to call Barr to testify before his panel.

Separately, Barr is scheduled to testify before a House Appropriations subcommittee on April 9, a House Appropriations spokesman confirmed Monday.

This story has been updated with additional developments Monday.