By Eric Bradner

CNN

WASHINGTON (CNN) -- After a chaotic week for President Donald Trump and his party, Republicans are facing a fundamental question: What's next?

With a new chief of staff, communications director and press secretary in tow, the President clearly is seeking to "hit the reset button," as his now former chief of staff Reince Priebus described it to Wolf Blitzer on Friday.

Also clamoring to turn the page, GOP leaders on both sides of Capitol Hill are clearly signaling they are ready to turn to a slew of issues and agenda items this fall, like taxes, border security, the budget, spending bills, and more -- and even Trump administration officials are echoing that call.

There may still be more moves to come on the personnel front, but on the agenda front -- despite Senate Majority Leader Mitch McConnell declaring in the early hours of Friday morning that it is "time to move on" after the failed vote on health care -- the President hardly sounds ready for a reset.

"Unless the Republican Senators are total quitters, Repeal & Replace is not dead! Demand another vote before voting on any other bill!" he tweeted Saturday afternoon.

Asked Sunday morning if that statement represents official White House policy, one top Trump adviser offered said "yes."

"Well, I think -- yes. And I think what you're seeing there is the President simply reflecting the mood of the people. Go and poll the American public and find out what the most important issue is to them right now, and it's health care," said Office of Management and Budget Director Mick Mulvaney told Jake Tapper on CNN's "State of the Union."

Mulvaney added, "And I think that's not only official White House position on this right now, it's sort of the national attitude towards it.

Another Trump confidant, former campaign manager Corey Lewandowski, made clear that the White House isn't letting go of the quest to repeal and replace Obamacare, telling NBC's "Meet the Press" that "there's three things on the agenda. It's tax reform, it's building a wall on the Southern border, it's repeal and replace of Obamacare, which didn't get done."

But throughout the weekend, Trump seemed less focused on moving on to tax reform or to the border wall than on confronting the procedural hurdles that threaten those efforts in the Senate -- where Republicans can only afford to lose two of their 52 party members if they are to advance legislation that doesn't have any Democratic support.

Trump called for the elimination of the 60-vote requirement to break legislative filibusters in the Senate. That 60-vote threshold didn't come into play on health care because Republicans were working under budget reconciliation rules that require just a 51-vote majority.

However, McConnell -- cognizant that if Democrats retake the majority in future elections, they could more easily advance their own agenda items without GOP support -- has said he'd never eliminate the legislative filibuster, putting him at odds with Trump, who tweeted several times on the topic on Saturday.

"Republicans in the Senate will NEVER win if they don't go to a 51 vote majority NOW. They look like fools and are just wasting time," Trump tweeted, following up in subsequent tweet saying "8 Dems totally control the U.S. Senate. Many great Republican bills will never pass, like Kate's Law and complete Healthcare. Get smart!"

The Trump administration has more health care decisions to make.

Repeatedly, Trump has warned he would let Obamacare "implode" in hopes that voters would penalize Democrats in the 2018 midterms.

Among the decisions his administration must make: whether to stop government payments to insurance companies to help lower costs for lower-income policyholders under Obamacare and whether to continue enforcing the law's individual mandate to purchase insurance.

"If a new HealthCare Bill is not approved quickly, BAILOUTS for Insurance Companies and BAILOUTS for Members of Congress will end very soon!" Trump tweeted Saturday.

White House counselor Kellyanne Conway said on "Fox News Sunday" that Trump will make a decision this week on the government payments to insurance companies.

"He's going to make that decision this week," Conway said. "And that's a decision that only he can make."

Health and Human Services Secretary Tom Price declined on NBC to address the issue directly, citing ongoing litigation.

"What I can tell you though is that the court has made a decision that those payments were made illegally," Price said. "And that's working its way through the court system."

But Price pledged to enforce Obamacare despite his belief that it isn't working, saying twice, "Our job is to follow the law of the land."

In an interview on ABC's "This Week," Price addressed the individual mandate question, saying his department could still choose to waive the mandate, which requires people to have health insurance or face tax penalties.

"All things are on the table to try to help patients," Price said.

Sen. Susan Collins of Maine, one of the three Republicans who sunk the "skinny repeal" of Obamacare on Friday, said on CNN's "State of the Union" that health care reform should return to the regular committee process. She also called on the Trump administration not to undercut Obamacare.

"I certainly hope the administration does not do anything in the meantime to hasten that collapse," Collins said.

For Democrats, health care conversations could shift toward single-payer "Medicare for all" proposals such as the one Vermont Sen. Bernie Sanders is expected to introduce in September.

Already, Massachusetts Sen. Elizabeth Warren and New York Sen. Kirsten Gillibrand -- both seen as potential 2020 presidential candidates seeking to take on Trump -- have backed single-payer, government-funded insurance. Other key Democrats will also be expected to weigh in on such a policy shift.

In Congress, meanwhile, McConnell has delayed the start of the upcoming August recess until August 11.

That's not the case on the House side, and not everyone is happy about it. "It blows my mind that we're probably not going to be here in August," House Freedom Caucus Chairman Mark Meadows said Friday.

House GOP leaders told rank-and-file members Friday at a closed-door meeting that they were plowing full-speed ahead on the rest of the spending bills, the budget and tax reform when the House returns to work in September.

Treasury Secretary Steven Mnuchin and National Economic Council Director Gary Cohn, along with key GOP leaders in the House and Senate, issued a statement Thursday afternoon laying out broad proposals and saying "the time has arrived" for relevant congressional committees to start drafting legislation.

As the Senate stays in session during the first two weeks of the traditional August recess, they are expected to tackle a long list of nominations to help fill out the Trump administration.

A key nomination awaiting further action is that of Christopher Wray for FBI director. Wray cleared the Senate judiciary committee with unanimous approval, which sets up a vote for Wray by the full Senate that is expected to be bipartisan -- something that's been hard to come by in the first six months of the Trump administration.

CNN's Eli Watkins, Ashley Killough, Deirdre Walsh and Ted Barrett contributed to this report.