Is the inquirer considered sinful for publishing details about his ex-fiancée, her flaws, and her family's flaws on a public Facebook group for the purpose of seeking a fatwa, and what is the expiation for this mistake, if any?

It appears that what you did is not backbiting (gheebah) that is prohibited, if your intention was to seek a treatment for problems in dealing with others, and if the group did not know the woman. Nevertheless, it would have been better not to publicize the story and to seek advice privately from someone you trust.