In Persuasion, why doesn't Anne just leave her family and go live with Lady Russell?
This is a great question - I have often thought the same! It is clear from the very first chapter that Anne is mistreated abominably by her father and sisters. Note what is said about her (from her father's perspective):
...but Anne, with an elegance of mind and sweetness of character, which must have placed her high with any people of real understanding, was nobody with either father or sister: her word had no weight; her convenience was always to give way; - she was only Anne.
Unfortunately, however, in spite of her poor standing and the way she is treated, we can also infer that Anne feels a sense of obligation in terms of trying to curb the excesses of her sister and father and also for looking after them. It would have been very strange for Anne to live with someone else and perhaps would have looked bad.