Hi Hope,
You're not unfixable! I've struggled with a similar need to contact my t excessively and want more out of the relationship than she could give. There were many times I felt sure that I could never get past the attachment issues! I also considered quitting more than a couple of times. But you know what? Even when I didn't think I could go another step, that I was broken beyond repair, I took another step forward. Then another step. Now, I can look back and see how far I've come. I'm in no way healed, but I have made a lot of progress. If I can do it, I believe you can too!
Attachment issues are painful!! There's no way around it! It takes time to work through them. But to work through them, we need to accept that the going isn't always going to feel good with our t. It is often going to feel painful. But by sticking with the work, and not leaving therapy when it gets painful, we can learn what is really behind that pain, where it came from...and then begin to take steps to heal that pain.
It sounds like your t is very willing to keep working with you toward healing. I know it feels bad having so many boundaries in the relationship, but those boundaries benefit you. The boundaries aren't there just to protect your t's private time (although that's part of the reason). The boundaries also help you learn balance in relationships. Many of us here didn't learn that growing up. We don't always know what's OK or not OK in relationships, or how much is "too much." But our t's can help us learn to trust and hang onto the therapy relationship, while still keeping it balanced.
You probably are not going to be able to stop contacting your t cold turkey. I know I couldn't. But why not sit down with him and ask him to help you develop a plan together for cutting down emails and phone calls. Would it help to have him tell you exactly how many emails he will accept per week? Then you can work on meeting that request.
Is there something the two of you could agree to together that you could do between sessions when the urge to contact him becomes too strong? It could be journaling, it could be going for a walk, praying, calling a friend, etc.
You might also need his help to come up with a plan for what you can do when you feel so hurt that you are tempted to quit.
Please don't give up! I know this is very hard, painful work, but you can do it!
PS - The only thing I disagreed with your therapist about was his statement that he doesn't need to prove to you that he cares or that he feels anything. I know that, in my case, my therapist DID have to prove that she cared about me. She didn't always do it in the way I wanted her to, such as reply to my email right away and in depth, but she showed it in other ways that I could see, feel, and trust. Without a way to know that my t truly cared about me as a person, I would have never trusted her enough to share my deepest thoughts and pains.