I would move on since she doesn't have what you're looking for. If the therapist I see told me she was taking a week off for self care, I'd assume something terrible had happened. Otherwise, I'd expect her to call it "vacation" like a normal person and not some special snowflake. I've also heard of some therapists taking every x week off, so maybe she's one of those. Of course, they have every right to do that, but it's something I would want to know up front (so I could find someone else). If you decide you want to stick it out with this one, you might want to ask about that.

The one I see seems to get her self care in by traveling on the weekends, which sometimes means she skips a Friday or Monday here and there. But I still get my sessions. I have two per week. If she's there at least three days during the week, I can expect to have both sessions. They are not at the same time every time. I don't know if that would be possible. It doesn't bother me personally since she isn't like the therapist you describe. I've never lost out on a single session because there wasn't room for me. Sometimes we've forgotten to schedule ahead of time and then she has just saved some slots around the time I usually see her. I am also in school, so my schedule and when I can see her changes every semester anyway.

Also, if you're going for psychoanalysis, this doesn't sound remotely like it. Is that what she claimed to practice?