That sounds very tough, but transference is nothing to be ashamed of, and there wouldn't be a specific name for it if the frequency with which it occurred hadn't required one. The therapeutic relationship is indeed strange and asymmetric. These feelings of wanting to maintain closeness with someone who you feel knows the inner you rather than that which you may feel you present to others, and still displays care towards you, are probably even expected to emerge. You don't know right now that she will tell you not to contact her – rather she probably considers it more beneficial for you to express your feelings inside the therapy room rather than from behind a computer screen. And yes, you pay for your sessions, but that doesn't mean that the relationship isn't real and that she doesn't care. If she was only worried about making money, she would have gone into city finance