(CNN) Emboldened by Donald Trump's surprise victory, Republican lawmakers vowed after Election Night that they wouldn't waste a single day to pursue their most urgent mission: Kill Obamacare.

The President-elect, too, has been in a hurry, declaring that the Affordable Care Act must be repealed and replaced more or less at the same time. "Probably the same day, could be the same hour," Trump said at a news conference last week

But on Capitol Hill, urgency doesn't always translate into speed.

In the first two weeks of the new year, Republican leaders have confronted resistance from rank-and-file lawmakers from across the ideological spectrum. Nervous about the potential political fallout of moving too fast on Obamacare repeal, some Republicans are now cautioning restraint -- a stark contrast from their ferocious attacks against President Barack Obama's signature health care law over the past several years.

Trump further complicated matters over the weekend when he told The Washington Post that he was nearly finished with his own plan to replace Obamacare and warned that he won't let Congress get in his way.

"The Congress can't get cold feet because the people will not let that happen," Trump said.

Although he didn't offer much in the way of details, the incoming president's comment that he wants both "insurance for everybody" with "much lower deductibles" sets the stage for a potential clash and drawn out negotiations with Republican lawmakers who have been stressing universal access over universal coverage.

"Repeal and replace is a bloody complicated exercise," said Brian Fortune, president of the Farragut Square Group, a health care consulting firm in Washington. "The challenge for the Republicans is, of course, to rework something without getting blamed for all the downstream effects -- quite difficult.

Senate Republicans got the ball rolling on Day One of the new session of Congress, introducing a budget resolution whose sole purpose was to repeal the health care law. Following Senate and House approval of the resolution last week, Republicans now get to work on crafting a second measure -- a budget reconciliation bill -- that contains the language to roll back big chunks of Obamacare.

But leading up to last week's first procedural vote, Republicans in both chambers expressed deep reservations.

The overarching worry among Republicans is that the party will vote to dismantle major portions of a law that covers some 20 million people before there is even a blueprint for an alternative, and be held responsible for widespread disruptions in health insurance.

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Complicating matters is the reality that several main pillars of the law enjoy broad support, such as the pre-existing conditions rule as well as a provision that allows children under the age of 26 to stay on their parents' health policy.

In the House, a mix of conservative and moderate lawmakers was skittish about supporting the budget resolution . They pressed House Speaker Paul Ryan for reassurances that the process of replacing Obamacare would unfold around the same time as the vote to repeal it.

GOP Rep. Charlie Dent of Pennsylvania told CNN before the vote on Friday that things were moving too quickly.

"I'm very concerned on the policy side specifically, that the replacement occur either simultaneously or as close to simultaneously as possible," Dent said. "If we don't provide a credible replacement plan, my main concern is that there would not be gaps in coverage for people who are currently subsidized. Also concerned about how the insurance markets might react."

Dent -- along with eight other House Republicans -- ultimately voted against the budget resolution.

And in the Senate, a group of five GOP senators introduced an amendment to the budget resolution to extend the deadline by which to craft the reconciliation bill -- the measure that would repeal Obamacare -- from January 27 to March 3.

While that deadline is largely viewed as symbolic and unenforced, the senators were sending a clear message: We need more time to figure out what's next.

"We just want to make sure that we get it right," GOP Sen. Susan Collins of Maine, one of the supporters of the amendment, told CNN. "One of the problems with Obamacare is that it was rushed through without input from Republicans for the most part, and we realize that insurance markets are complicated. And we don't want people to fall through the cracks."

Asked whether the amendment signaled that Republicans were increasingly less concerned about overhauling Obamacare on the most expedient timeline, Republican Sen. Bill Cassidy of Louisiana, another author of the amendment, pushed back.

"My gosh. If you're going to say we're going to fly to the moon but we're going to do it in March instead of next week, would you still feel it's pretty urgent?" Cassidy said. "You'd say, oh my gosh, we've got to get to work!"

Under pressure, Ryan has grown increasingly emphatic that there will not be long lag time between a repeal vote and when Congress considers a replacement package or set of measures.

At a CNN town hall last week, Ryan went as far as to say that GOP leaders would repeal and replace "at the same time." But other than to say that there would be action within the first 100 days of the Trump administration, he declined to commit to a specific timeline, noting that the whole process would take "a little bit of time."

But even while lawmakers in his party are growing wary of acting too quickly, Trump has been ratcheting up the pressure on Republicans to help him deliver on his campaign promise as fast as possible.

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And if in fact Trump does release an Obamacare replacement plan of his own, that could present a series of new challenges for Ryan and Senate Majority Leader Mitch McConnell who are under pressure to satisfy an array of ideological priorities from rank-and-file members.

At his news conference in New York City last week, Trump said that a plan to repeal and replace the health care law would be submitted "as soon as" Tom Price, his nominee for Health and Human Services secretary, is confirmed. Price, a Georgia congressman, will testify before the Senate Health committee on Wednesday, but the Senate Finance committee, which will vote on his confirmation, has not yet set a hearing date.

It's unclear what role Price will have in shaping the GOP's new healthcare system, and a Trump transition official told CNN that the incoming administration is "taking nothing for granted" before the congressman's confirmation.

"It's all about the confirmation," the official said. "Nobody's looking past that date."