There are several reasons:
First, Anne is a single woman. No unmarried woman is supposed to leave their household unless they are married. They are supposed to go from their father's house to their husband's house- that was just custom.
Second, Anne was all about self-sacrifice. Even if she had the opportunity (which she did) to beg Lady Russell to take her away from her crazy household, she would have not done it out of respect and pity for her father. Instead, she worked out a plan for her Dad to help him out in cutting expenses and living more modestly. To this, of course, he turned his nose.
Third, it would have made her father even more ashamed and will have made him more aware of his new status if she had left to leave due to family ruin. No, the Elliots cannot shake anything nor move any pieces that may point out that things are changing. Therefore, it was best to leave things as they were, and make the best of it.