At the height of the pandemic, Gov. Andrew M. Cuomo called upon some of his most trusted emissaries to return to the fold to help coordinate the state’s coronavirus response, including Larry Schwartz, his former top aide who is now leading New York’s vaccination efforts. But with Mr. Cuomo facing concurrent scandals and calls for his resignation, Mr. Schwarz has taken on another role: as a political operative, asking state Democratic leaders to support the governor, a third-term Democrat, while continuing to discuss the urgent business of immunization. According to two Democratic county executives, Mr. Schwartz placed calls to them...