HAUPPAUGE, N.Y. — A recent day in May began in crisis mode for Steve Levy, the Suffolk County executive, over an allegation that a police crackdown on unlicensed drivers amounted to the ethnic profiling of illegal immigrants.Soon, he was preparing a response to accusations by 30 state legislators, fellow Democrats all, that his policies in this Long Island county of 1.4 million “instigate divisiveness, hatred and intolerant behavior toward immigrants.” By 2 p.m., Mr. Levy seemed relieved to turn from “the whole illegal immigration thing,” as he calls it, to a topic confined to his 631 area code: a health...