AVA: My Story By Ava Gardner (Bantam: $19.95; 315 pp.). The nicest thing ever said about her, writes Ava Gardner, was George Cukor's calling her "a gentleman." It's easy to see why she liked the term; though Ava often drank into the wee hours, and addresses her readers as "honey," she never traded in her North Carolina farm-girl soul for the Hollywood model. Stardom, she writes, "gave me everything I never wanted," and she admits she wasn't a great actress despite good notices for films such as "The Sun Also Rises" and "The Night of the Iguana." And her love life indeed proves as interesting as her career, boasting an all-star cast; her three husbands were Mickey Rooney, bandleader Artie Shaw and Frank Sinatra. None of her bed partners comes off very well, which suits the mood of this autobiography (assisted by Alan Burgess and interim Los Angeles Times book editor Kenneth Turan): lighthearted, but with an undercurrent of sadness and resigned dissatisfaction.