Investigators have interviewed at least three Democratic county executives who said they were surprised to receive calls from Larry Schwartz, a volunteer adviser who oversaw vaccine distribution for the state, asking whether they would be calling for Mr. Cuomo’s resignation, the people said. Mr. Schwartz contacted more than a half-dozen executives in early March, executives said, as Mr. Cuomo faced mounting calls for his resignation from members of his own political party and was personally contacting state lawmakers to shore up support. The executives said that at the time of Mr. Schwartz’s calls they were typically speaking with him about the allocation of vaccine supplies, not politics. Mr. Schwartz has said he didn’t link vaccine distribution to political considerations. On Friday, he referred questions to his lawyer, Guy Petrillo, who declined to comment.