Moderate Republicans urged the party to “move on” from its fixations on immigration reform and Benghazi on Sunday, as Barack Obama continued to defend his decision to circumvent Congress by use of executive action.

The president’s controversial decision on Thursday to grant legal status to 5 million undocumented immigrants has provoked a storm of protest from opponents who claim he overstepped his executive authority.

But Sunday’s political talk shows brought relatively few suggestions for overturning the decision from Republicans, who are also reeling from a report slipped out by the House intelligence committee on Friday that largely exonerated the administration’s handling of the 2012 Benghazi terrorist attack – another once-reliable target for criticising Democrats.

The Arizona Senator Jeff Flake, one of eight Republicans and Democrats who were behind a bipartisan immigration bill in 2013, urged a fresh attempt to tackle issues such as border security, guest workers and law enforcement rather than seeking to overturn Obama’s action on legalising existing immigrants.

“He’s done one portion of the latter. So I’d rather move legislation on the other three items and put it on his desk,” Flake told NBC, though he accepted this would now be more difficult.

“We did much of what the president did, and, in fact, went even further in the Senate bill,” he added of the 2013 bill, which passed the Senate but was never voted on in the House. “The problem is, the way he did it is going to make it very difficult to move the other parts of immigration reform that we really need. So it’s not that he did something that we wouldn’t have done otherwise; it’s the way he went about it.”

Lindsey Graham of South Carolina, another Republican member of the Senate’s original “gang of eight”, also rejected calls from some conservatives to impeach Obama or risk shutting down the government by blocking funding for the measures.

Instead, he focused in an interview with CNN on the “dangerous” precedent of using executive action to overcome frustration with Congress and claimed Obama had “made a political decision to try to divide the Republican party”.

Even the former senator and presidential candidate and conservative firebrand Rick Santorum called for a “ratcheting” response to the president on immigration, rather than reacting more combatively. He too warned CNN that Obama had “opened up Pandora’s box for every president in future … to create new law”.

In an interview on ABC, Obama defended his decision on the grounds that presidents often “exercise prosecutorial discretion”, and claimed Republicans risked stalemate if they tried to overturn his order.

“There’s often a lot of rhetoric coming out of Congress, and in Washington. But it doesn’t match up to what I think the American people expect,” he said.

“One of the habits that we’ve seen in Congress over the last four years since the House Republicans took over is that everything becomes hostage to one disagreement. So a couple of years ago it was Obamacare, the Affordable Care Act. And they decided, ‘We’re going to shut down the government because we disagree with this one law.’”

Though opinion polls show widespread unease about relaxing immigration laws, Obama’s move nonetheless presents a tactical challenge for Republican leaders, who are looking to use their newfound control of Congress to impose their own policy ideas rather than to simply oppose the White House.

Flake urged the party to drop its focus on the administration’s fumbled response to the killing of the ambassador to Libya, Chris Stevens, in September 2012, following the Republican-led intelligence committee report on Friday which concluded there was no evidence of a deliberate coverup.

“I always thought the biggest problem with Benghazi is how it was cast by the administration, and the remarks of Susan Rice just really flew in the face of what we knew was going on,” Flake told NBC. “But with regard to other things that were addressed by this report, yes, I’ve thought that for a long time we ought to move beyond that.”

Senator Graham, a noted foreign policy hawk, was more scathing, calling the House report “crap” and urging fresh attempts to investigate the affair which many see as a indictment of then secretary of state, and likely 2016 presidential candidate, Hillary Clinton.