The new chairman of President Reagan's troubled AIDS advisory commission appointed a new executive director to lead the panel's professional staff. Retired Adm. James D. Watkins named Polly L. Gault, who served as staff director of the Senate subcommittee on education, arts and humanities, to do the staff hiring and direct the day-to-day operations of the AIDS commission. He said the appointment marks "a new stage of heightened productivity for the commission." Gault replaces Linda D. Sheaffer, who was dismissed last month because of staff leadership problems. Last week, the panel's chairman, Dr. W. Eugene Mayberry, and vice chairman, Dr. Woodrow A. Myers Jr., resigned.