(CNN) - Heads "should have rolled" at the White House over the botched kickoff of the HealthCare.gov website, said Rep. Nick Rahall, who also is skeptical of President Barack Obama's proposal to fix another problem resulting from the rollout of Obamacare.
Rahall, who is regularly targeted by Republicans in his West Virginia swing district, told reporters on Thursday that the health care debacle is hurting him in his reelection bid.
He said outside groups are running ads against him on the President's previous pledge that people who liked their health plans could keep them under Obamacare, which has not proven true.
"I'm going to have to respond," he said.
Obama has apologized for his broken promise and announced steps on Thursday to fix the cancellation problem administratively.
He also acknowledged that the rocky rollout of the Obamacare website beginning October 1 and the cancellation issue has put a burden on Democrats whether they are up for reelection or not.
Some Democrats appeared to be breathing a little easier on Thursday following a meeting with White House Chief of Staff Denis McDonough, who went over details of steps to fix the website and address the cancellation issue.
A meeting with another administration official on Wednesday was much more contentious.
The Affordable Care Act, Obama's signature legislative achievement, passed Congress in 2010 with only Democratic backing.
Republicans, especially in the House, have since sought to derail the sweeping health care reform measure, and have seized on its recent rollout problems as proof that it's unworkable.
Rahall is reviewing a Republican proposal that would allow those with canceled plans to extend them into next year. It would also gut a major part of Obamacare by permitting anyone to purchase them, even though their existing policies don't meet the tougher requirements of the Obamacare initiative.
The House is set to vote on the plan on Friday.
House Democrats are planning to unveil their own proposal.
Asked if it helps Rahall at home that Obama went out of his way to try to take the blame for the health law's shortcomings, Rahall said "not really."
"It's been on him all along and look at where we are," he added.
Asked if the President's proposal gives Democrats in 2014 enough cover, McDonough told reporters, "This isn't about elections."
"This is about making sure people have affordable health care," he added.
One member said the mood at the meeting with McDonough was "vastly improved" from Wednesday's session with a senior Obamacare adviser, that they got answers to key questions this time.
At one point, members applauded McDonough's presentation on the website and cancellation fixes.
Questions focused on the proposed policy solution, with several pressing over how it would work in their states.
House Minority Leader Nancy Pelosi solicited input on several different ideas for the Democratic policy remedy. One involved some direction to state insurance commissioners.
Rep. Elijah Cummings, D-Maryland, said he thought Obama's plan on policy cancellations was sufficient, and that McDonough told them the White House had talked to the insurance industry about it and is not worried about industry being able to handle any changes.
- CNN's Lisa Desjardins contributed to this report.