I couldn't see Anne as so lacking in self-esteem as to not accept an offer after the others have refused it. I can't point to any other situations where Anne stands up for herself, but she is certainly fearless enough in standing up for other people (e.g. her advice to Elizabeth about Mrs Clay, ineffective though it was). But it seemed to be the only explanation - until you pointed out the Captain Wentworth's sister and brother-in-law effect.