When a loyal leader on your own team says there is a “crisis of confidence” surrounding your signature initiative, you’ve got trouble.

That’s the phrase Democratic Sen. Barbara A. Mikulski of Maryland used repeatedly Tuesday morning to describe the rollout of the new health care law as she questioned Marilyn Tavenner, the head of the health agency tasked with overseeing the law’s implementation.

“I believe that there’s been a crisis of confidence created in the dysfunctional nature of the website, the canceling of policies, and sticker shock from some people,” said Mikulski, who has generally been a strong ally of the administration.

She cited a news report that 73,000 people in her own state are getting cancellation notices, “so there has been fear, doubt and a crisis of confidence” — and she’s worried people, particularly the young, won’t enroll as a result.

Indeed, the Affordable Care Act’s website woes, combined with millions facing the cancellation of their individual policies despite President Barack Obama’s assurances that would not happen, have put Hill Democrats in an increasingly awkward position — with no easy way out.

After celebrating their victory over the GOP during the government shutdown by sticking firm on Obamacare, vulnerable Hill Democrats are now looking for cover.

The White House now is in a race to fix the problems before demands within the Democratic Party for legislative fixes become overwhelming — either in the form of a proposal by Sen. Joe Manchin III of West Virginia for a one-year delay in the individual mandate or Louisiana Sen. Mary L. Landrieu’s new proposal to keep grandfathered plans online instead of generating more cancellation notices.

Landrieu, of course, is up for re-election in 2014, and it’s no secret the GOP intends to use the health care law’s troubles as fuel to defeat her and other vulnerable Democrats.

Senate Minority Leader Mitch McConnell called the developments “foxhole conversions” by Democrats. “I think we’re witnessing the beginning of a stampede away from the president’s signature accomplishment,” the Kentucky Republican said.

“I think what’ll be really interesting to see in the Senate is the number of Democrats in very red states who are up in ’14 and what they start demanding of the majority leader and the administration, in terms of adjustments to this law,” McConnell said. “We all know they were lockstep a couple of weeks ago — everybody voting against defunding, voting against delaying and all the rest. Now, it seems to me we’re hearing a kind of different tune.”

One such Democrat, Sen. Mark Begich of Alaska, said he’s going to look at Landrieu’s proposal as well as legislative fixes of his own.

“We have a couple in our office we’re kicking around but nothing to make public yet,” Begich said. “Whatever we can do to make sure people have insurance, we should be focused on. I’ve always said the whole law is not perfect, and you should always look for tweaks. I’m not afraid to tweak the law.”

The White House and Senate Majority Leader Harry Reid may be able to fend off meaningful defections for now. For example, Reid brushed off a question about allowing a vote on Landrieu’s bill Tuesday and dismissed McConnell’s remarks by saying the minority leader should worry about his fellow Republicans.

But if the administration can’t fix HealthCare.gov by the end of November as promised, there will be enormous pressure to do something. And apart from the mechanics of the law, the White House continues to struggle to explain how a key promise made by the president — that people would be able to keep their plans, “period” — isn’t coming true.

Press Secretary Jay Carney on Tuesday blamed insurance companies for not continuing to offer the grandfathered plans, saying that they had pulled the rug out from under people.

The White House has not endorsed any legislative tweaks aimed at restoring those plans — and warned if some were to pass, they could undermine the broader law.

“If you are going to assert that insurance companies can continue to offer substandard plans, bad-apple plans for example, that do not provide hospitalization or have carve-outs that exempt from coverage the very chronic condition you may have, often in a way that the purchasers of this insurance don’t even know … that undermines the fundamental promise of the Affordable Care Act, which is that everyone in America should have access to affordable, quality health care coverage,” Carney said.

And, of course, in speech after speech — until Obama added the caveat about insurance companies Monday night — the president had been emphatic. On Tuesday, one of Carney’s many attempts to explain away the president’s words went like this: “He didn’t say … ‘if your insurance company cancels your plan and gives you something else that’s worse, you can keep it.’ He said that ‘if you had a plan — if you have a plan that you like, you can keep it.’”

Carney said the people affected were only a portion of the 5 percent of the population in the individual health insurance market. But that is still millions of people who are being told now that they can’t keep their plans, and it’s causing an enormous political headache on Capitol Hill.

Emma Dumain contributed to this report.