Question: Even after Jesse tells Carrie straightforward that he was engaged, why would she still insist that he keep the promise he made to her ten years prior? Anyone else would've simply put the past in the past and moved on, especially after learning that their former lover with whom they lost touch is about to marry another person (so as to not arouse any suspicions).
Answer: I agree with the other response. I've heard of people trying to get an ex back, people wanting a friend to end a relationship/marriage and be with them instead, etc. Not everyone would "simply" move on.
Answer: There's several people out there who would try to get their ex back before it's "too late" in a sense before they get married. This is often a plot line in movies for a reason. Doesn't mean it's right of course, but there are plenty of people out there who would attempt to coax an ex into breaking off an engagement for hopes of a second chance whether it's right or not.