Is the questioner sinful for continuing to be called by the name "Iman" despite her knowledge of the fatwa prohibiting naming with it, her inability to officially change it, and her family's refusal to call her by another name?

Among the Prophet's guidance (peace and blessings be upon him) was to change names that imply self-praise for their bearer, such as the name "Barrah" (meaning pious/dutiful) which the Prophet (peace and blessings be upon him) changed to "Zainab," because it involved a commendation of the self. Accordingly, the name "Iman" (meaning faith) carries a commendation of the named person, and it should be changed. However, if you are unable to change it, there is no harm, because the prohibition is due to dislike (karahah), and a disliked matter is waived by the slightest need.