Republicans and Democrats share at least one sentiment on healthcare: Someone will bear the Obamacare blame, and it's not gonna be them.

The Capitol Hill ground game on Obamacare repeal and replace began in earnest Wednesday, as President Obama and Vice President-elect Mike Pence rallied their respective parties to stand their ground in the healthcare fight.

Later that day, the Senate started a budgeting process that Congress will use to wipe out big parts of the healthcare law. And as Republican leaders lobbed sharp criticisms at the law and promised a seamless transition to something better, Democratic officials rolled out a new attack line: Republicans are making "America sick again."

"Instead of fulfilling their promise to 'repeal and replace' the Affordable Care Act, Republicans are going to make America sick again by offering no health care plan to the American people," says a memo sent to the rank-and-file, a riff off President-elect Trump's campaign slogan.

Both sides are acutely aware that the individual market health insurance situation is delicate, and Americans will want someone to blame if their premiums keep rising or if they lose their plans.

Republicans in Congress are moving forward on their plans to repeal much of the healthcare law using the budget reconciliation process. But that will take a few weeks, and crafting its replacement will take months at the very least.

So the game right now is to shift the blame to the other party. Democrats are warning that Republicans will strip away coverage for Americans, and Republicans argue that the law was rotten to the core thanks to Democrats.

"The simple fact is the American people know who owns Obamacare," Pence told reporters. "It's the first half of the title. It is Obama and the party of Obama."

House Ways and Means Chairman Kevin Brady said "an effective transition is really the key." Rep. Chris Collins, R-N.Y., said people need to know Republicans aren't going to "pull the rug out from under anyone."

Pence emphasized that Trump and his White House staff are already working toward an "orderly transition" out of Obamacare. And Trump himself tweeted some warnings to Republicans as they met with Pence Wednesday morning.

"Republicans must be careful in that the Dems own the failed Obamacare disaster, with its poor coverage and massive premium increases …" Trump tweeted.

Most people are relatively shielded from the direct effects of Obamacare repeal because they have health plans through their employers. But coverage for about 21 million people is at stake, for the approximately 10 million people with marketplace coverage and 11 million newly covered by Medicaid.

While Republicans are going to great lengths to stress that they won't leave those people hanging, Democrats are casting doubt on whether they will be able to agree on a replacement. They never passed a plan during the last seven years of attacking Obamacare, House Minority Whip Steny Hoyer noted.

"Where have they been?" Hoyer said. "Where they have been is on a rhetorical binge demagoguing a piece of legislation that they don't have an alternative for. They like the rhetoric; they don't like the reality."

"It's their responsibility, plain and simple," said Senate Minority Leader Chuck Schumer.

The goal for Republicans is to pass an Obamacare repeal bill before the spring, possibly in February, and craft a replacement through the committee hearing process. Pence didn't lay out a specific deadline for when a replacement should be ready, or how long the transition should last. Aides and lobbyists say there's been no final decision made about how long to keep Obamacare intact for a transition period.

But there's a strong sense that a replacement bill should be ready within six months or so, Collins said. "Within six months we should be able to put pen to paper and have a replacement plan," he said.

Things will get more difficult for Republicans at that point, because they will need buy-ins from some Senate Democrats to get new reforms passed. And Democratic leaders have said they won't join a dismantling of Obama's signature domestic reform.

Senate Majority Whip John Cornyn insisted Republicans will at least win the messaging wars if they keep their promises to produce an Obamacare replacement. Schumer and other Democrats are wishing for them to fail for political reasons, Cornyn said.

"I understand [Schumer's] political argument," Cornyn told reporters. "He is praying and hoping for failure, which means he's praying and hoping for more pain on the part of the American people."

Susan Ferrechio and Nicole Duran contributed to this story.