Please KNOW that I am NOT judging......But it may be possible to rehome her. It has been done before, successfully. The people in my rescue group, who have MANY more years experience than I do, have told me of stories with happy endings, from situations just like yours. My first question would be "are there any other no-kill rescue groups in your area?" And don't look only in "driving distance". because many groups will help get the cat to them if it is further away. All you need to do is ASK around.
I do want to say I am sorry you have to go through this, it is difficult, I know.