The Assembly investigation — and a parallel inquiry overseen by the state attorney general, Letitia James — could take months to complete, effectively buying the governor time to repair his battered public image.

And the governor, his supporters and his aides have, in fact, worked to shore up support behind the scenes. A Democratic political operative who has been a longtime ally of the governor, Charlie King, has in recent days solicited public comments urging that the investigations be given time to be completed.

For most of his tenure, Mr. Cuomo has relied on a close set of advisers who act as both political enforcers and point people on government operations. In February, The New York Times reported that at least nine top officials in the state Health Department had resigned or retired during the pandemic as Cuomo aides acted without their input or expertise.

Mr. Cuomo had made it clear, in public and private comments, that he believed state public health officials had no understanding of how to conduct a large-scale operation like vaccinations, and that his close aides, who like Mr. Schwartz did not have public health experience, could do a better job.

Mr. Schwartz, who is unpaid, has been in charge of vaccination planning, a position that puts him in frequent contact with local officials.

Indeed, the two county executives said they spoke on the condition of anonymity because they did not want to endanger their local vaccination efforts.

The phone calls by Mr. Schwartz, who for much of the pandemic has lived with Mr. Cuomo in the governor’s mansion, put on stark display the political concern for Mr. Cuomo, who has long held a dominant place in state politics, and just a year ago was well on his way to becoming a hero of the pandemic, with soaring approval ratings.