WASHINGTON – Still sore after the whipping administered by President Obama and Senate Democrats over the government shutdown, congressional Republicans nonetheless insist they remain committed to shutting down the Affordable Care Act.

House Republicans, especially those with ties to the Tea Party, targeted what is popularly known as Obamacare from the outset of the raucous debate, refusing to support a temporary funding measure to carry the government beyond the federal fiscal year that began on Oct. 1 unless the healthcare law was defunded.

That effort famously went up in flames when lawmakers finally embraced a settlement that reopened the doors after a little more than two weeks without killing Obamacare. Now a question is plaguing a divided GOP caucus – how does the party accomplish the commonly held objective of bringing the ACA to a quick and bloody end since the Senate as presently composed would never go along? And, even if the upper chamber agreed, the president would issue a quick veto that could never gain the votes to be overturned.

In addition, Republicans must find some way to recover from the perception that the showdown over the shutdown was their Battle of the Bulge in the Obamacare fight – that last great shot at victory.

“This is our last chance and our last best chance to do something about this,” Sen. Marco Rubio (R-Fla.) said back in July.

But those long odds aren’t stopping congressional Republicans, particularly conservatives, from ramping up the fight. Some, like Sen. Ted Cruz (R-Texas), argue that the law’s bitter critics will do whatever it takes.

Cruz said he is “optimistic” about the ultimate death of Obamacare.

“I am inspired by the millions of Americans who have risen up,” he said. “And if the American people continue to rise up, I am confident that in time the U.S. Senate will follow the lead of the House of Representatives and listen to the American people. That is our job. That is our responsibility.”

A public revolt, he said, will “turn this around.”

“We’re going to restore jobs, we’re going to restore economic growth, we’re going to restore the ability of people struggling to climb the ladder and achieve the American dream and we’re going to stop the number-one job killer in this country that is Obamacare,” he promised.

Appearing a convention of the Texas Medical Association in Austin on Saturday, Cruz said the cause was lost because “Senate Republicans declined to unify and declined to support House Republicans” in repealing the law. He remains “hopeful that in the future the Senate will listen.”

Cruz has not ruled out navigating toward another governmental shutdown to push the issue when the current continuing resolution expires on Jan. 15. But apparently he will have to maneuver around Senate Republican Leader Mitch McConnell, of Kentucky, if he intends to do so.

Like Cruz, McConnell wants to kill Obamacare.

“This law is ravaging our economy, killing jobs, driving up premiums, and driving people off the health care plans they have and like, in droves,” McConnell said. “Its disastrous rollout is sign of even worse things to come. And the Democrat refusal to delay it reflects a stubborn ideological obsession that will do untold damage to our country. And Republicans remain determined to repeal this terrible law.”

But they will have to accomplish that, he said, without closing the government a second time. Appearing on Face the Nation on Sunday, McConnell said, “There will not be another government shutdown — you can count on that.”

“A number of us were saying back in July that this strategy could not and would not work,” McConnell said. “And of course it didn’t.”

McConnell acknowledged that Senate Republicans “have a math problem” when it comes to getting rid of Obamacare.

“It’s the following math problem — 55 Democrats and 45 Republicans,” he said. “We only control a portion of the government, and so that limits our ability to get rid of this horrible law.”

Sen. Lamar Alexander (R-Tenn.) concurs with McConnell, asserting that another government shutdown could prove disastrous.

“Every one of us has voted against Obamacare repeatedly,” Alexander said of Republicans. “Every one of us would vote to repeal it. But in my view the right tactic is not to shut down the government. It won’t work – Obamacare would just keep going and we would have shut down the government.”

Alexander suggested one way to end Obamacare – “elect more Republicans.”

“The best way to do it is to take over the government,” Alexander said. “Elect some more senators. Elect a president. Put in a bill. That’s our constitutional system.”

Unlike either Cruz or McConnell, Alexander, the ranking member on the Senate Health, Education, Labor and Pensions Committee, has a plan for replacing Obamacare. It includes providing governors with more flexibility in operating state Medicaid programs, strengthening workplace wellness programs, permitting small businesses to pool their resources and offer lower-cost insurance plans for employees, expanding opportunities for consumers to purchase insurance across state lines and providing greater access to health savings accounts.

Alexander said his plan offers “step-by-step reforms that would reduce the costs of healthcare.”

Alexander isn’t alone in advocating for a process known as “repeal and replace.” The House Republican Study Committee, a conservative group within the GOP caucus, also has offered an Obamacare alternative that it intends to continue pursuing.

“American families and businesses deserve and demand real solutions to the serious problems that exist in our healthcare system,” said Rep. Steve Scalise (R-La.), the committee chairman. “The RSC’s American Health Care Reform Act is a common-sense bill that will lower costs using conservative, free-market solutions which give American families more choices without the unworkable mandates and billions in taxes included in President Obama’s healthcare law.”

The 200-page bill offers $20,000 in tax deductions to families and a $7,500 deduction to individuals to purchase insurance from vendors in any state — thus, supporters say, allowing people to save money by selecting lower-cost providers.

The measure also offers altered proposals to some of the more popular aspects of Obamacare – creation of a $25 billion fund to lower costs for those afflicted with pre-existing conditions, permitting people to carry their insurance from job to job and permitting coverage for adult children up to age 26.

“By allowing people to buy health insurance across state lines and pool together so small businesses and families can get the same buying power as large corporations, we can lower costs and increase choices while removing Washington bureaucrats from your healthcare decisions,” Scalise said.

But no one seems to have developed a strategy of repealing Obamacare and replacing it with a package of free market alternatives more to the liking of conservative lawmakers. Regardless, Rep. Paul Ryan (R-Wis.), chairman of the House Budget Committee and 2012 Republican nominee for vice president, urged that the fight go forward.

“The way I see it, our job is to preserve our values in the 21st century,” Ryan said during an appearance at the recent Values Voters Summit in Washington DC.

“We need to completely rethink government’s role in healthcare,” he said. “That means we can never give up on repealing and replacing Obamacare.”

In an op-ed that originally appeared in USA Today, Rep. John Fleming (R-La.), a family physician for more than 30 years before entering Congress, insisted that “Obamacare is just the latest bad law needing repeal.”

“That’s why we fought Obamacare through this difficult debate, and that’s why this is just one round and why we must continue to fight on,” he said.