Washington — President Reagan has chosen Elizabeth Dole to be his new secretary of transportation. The nomination of Mrs. Dole, who would replace Drew Lewis, must be confirmed by the Senate. Mrs. Dole, wife of Sen. Robert Dole (R) of Kansas, is currently serving as special assistant to the President for public liaison. She also worked for the Federal Trade Commission and served as deputy director of the President's Commission on Consumer Interests under former President Richard Nixon. She is the first woman named by Mr. Reagan to a cabinet post.