Something is not working correctly in this client-t partnership, because you remember and write here about the times she failed. I would expect to see, if therapy was working, a gradual improvement. But this is more like an extension of how your parents werent there for you. But you experience things as getting worse.
As for other clients being allowed to email or text their ts - my t offered to be available over the xmas holidays one year (in addition to our regular sessions). But that was his choice of time, his offer, and it was time-limited.
Maybe more frequent sessions would help? Do you leave feeling up or down? It helped my mood when i figured out it was okay to be happy arriving and leaving my sessions. It seems counterintuitive - we go to t because we are depressed - but actually we go to improve our lives, to have hope. Otherwise, when are we supposed to make this switch from sad to happy? I actually asked him this - then i started leaving happy. He used to annoy me coming to get me in the waiting room and he would be grinning like an idiot - but then my mother was never happy to see me. T is.
Hope this helps.