NEW YORK — With the full implementation of the Affordable Care Act rapidly approaching, efforts to enroll millions of Americans in new health plans are taking center stage in Washington and beyond.

The technical failures of the federally run health insurance website, healthcare.gov, have continued to draw the ire of Obama administration critics, who contend that the glitches are a sign that the new health system is unworkable. Yet in recent weeks the focus has turned to people whose current coverage is being canceled because it does not meet the law’s new requirements.

At a House Ways and Means Committee hearing in late October, lawmakers expressed their concerns to health care officials. “My constituents are frightened,” Rep. Kevin Brady (R., Texas) told Marilyn Tavenner, administrator for the Centers for Medicare & Medicaid Services, whose department oversaw the creation of President Obama’s Health Insurance Marketplace.

“They are being forced out of health care plans they like,” he said. “The clock is ticking. The federal website is broken. Their health care isn’t a glitch.”

Since state and federal health insurance exchanges opened on October 1, insurance companies have been sending notices to hundreds of thousands of people in the individual insurance market informing them that their existing plans will soon be canceled.

In most cases, the insured have been offered new plans, often with better coverage but also higher prices.

Like Brady, Republican lawmakers in both houses of Congress have seized upon the cancellations to criticize the Affordable Care Act and call for its repeal.

Political observers say the cancellation issue is a godsend for the GOP, which was becoming increasingly concerned that its narrowly focused criticism of the problem-plagued healthcare.gov could lead to a dead end once the website’s issues are addressed.

“They’ll fix the problems with the website,” Sen. Tom Coburn (R., Okla.) told The New York Times. “I think they won’t fix the problems with the bill.”

Republicans’ criticisms have angered the president and his fellow Democrats, who have been quick to point out that the last major GOP effort to change the country’s health care system —the 10-year-old Medicare Part D program — was also plagued by problems when it first went into effect.

Speaking in Massachusetts late last month — on the same day that Health and Human Services Secretary Kathleen Sebelius was being questioned by a congressional panel about the website problems and the policy cancellations — Obama explained that some people who have had what he called “substandard” insurance plans may have to choose another one now that the Affordable Care Act has gone into effect.

He accused lawmakers in Washington of distorting that fact by failing to mention that the new plans will be more comprehensive and often come with cheaper premiums.

“If you leave that stuff out, you are being grossly misleading, to say the least,” the president said.

Both Obama and Sebelius apologized for the technical failures that have prevented millions of people from signing up for health care coverage through the federal insurance exchange. They said that the troubles were being fixed and vowed that the rollout of the health care reform would be successful.

“We are going to see this through,” Obama said, pounding his fist on the podium during his Boston speech.