In Maryland, the turmoil around the state-run exchange is dogging Lt. Gov. Anthony G. Brown, a Democrat hoping to succeed Gov. Martin O’Malley, who is leaving office because of term limits. Mr. O’Malley made Mr. Brown his point person on the health care law’s rollout in Maryland, and now Mr. Brown’s opponents — both Democratic and Republican — are bludgeoning him with the state exchange’s failures.

Image Credit... Derek Montgomery for The New York Times

Addressing a legislative panel on Jan. 14, Maryland’s top health official said the state had made the mistake of using “off-the-shelf” software with serious defects. The official, Joshua M. Sharfstein, said the state would stick with the current system until the enrollment period ended on March 31, but might consider other options, such as joining the federal exchange, after that.

All five states lean Democratic, and many analysts believe the exchange failures will not be enough to allow Republican upsets this fall. Still, the issue has been potent enough to hurt Mr. Obama’s approval rating and put congressional Democrats on the defensive. If nothing else, it will provide a continuous headache for Democratic candidates like Mr. Dayton in Minnesota, who is hoping to cruise to a second four-year term. The website of the state’s exchange, known as MNsure, is still plagued by software defects, and an outside review last month found that it cannot be completely fixed before the March 31 enrollment deadline.

“The reality is that Mark Dayton owns the MNsure debacle — there’s no other way about it,” said Ben Golnik, a Republican strategist and a former executive director of the Minnesota Republican Party. “He pushed it through the Legislature, he said Minnesota is all in, so I think it’s going to be very difficult for him to distance himself from this.”