My advice may not be something you'd want to hear, but, I believe, it'd be better for you to find a new therapist, someone who is drastically different in their style from this one.

I don't know what this woman is triggering in you, but if this is something specific about her that you are reacting to, you need to understand what it is. In my experience, this is better done with somebody else, not with the same person who triggers those reactions. So, no, I don't believe that telling her about what you are doing will bring any therapeutic benefits, as well as staying in therapy with her. What is happening with you is a rather a big thing that needs to be understood in the context of therapy, but I don't believe this can be done with this particular therapist who triggers your obsession. This would be too much for you and for this therapist to do because it'd be too intense and too personal to make it therapeutic. This type of exploration is possible only in a much more emotionally cool and neutral space. So, you'd be better off with a different therapist at this point, I believe.