To be honest, I would move on. My first T was like this too. She would let me sit in silence for long periods of time fully knowing how much anxiety it was causing me. Even trying to discuss it with her, she would not change her 'technique'. She also was not into emailing or outside contact which at the time I assumed was what all therapists were like. I was really attached to her too, so much so that when we moved countries I made a couple of trips on my own to see her. When one day she randomly said she thought I might like to think about finishing therapy, I cried for 5 hours all the way home. Looking back it was really bad of her to have announced that when I was away from home and vulnerable and although I did feel attached and dependent, she just wasn't the right therapist for me at all.

I moved on and have been with my current T for nearly 10 years now and she is wonderful. She allows all the things my first T didn't and more importantly if I tell her something isn't working for me, she responds. She adapts. She does not let me sit in silence, or checks with me first if it's ok if she thinks I need some time to think. She is very open to feedback and we work together to find the right approach. All of this has helped me hugely. I could not have got any of this from my previous T and would have wasted alot of time in yearning for it.

I know it's really hard when you're attached but maybe consider how much your therapy might be improved if, say, you didn't have the silences anymore and you did get email responses and the things you feel you need. Maybe take a break and you may find you feel less attached after all. I would also consider as others have said, meeting with some other T's just to hear how they do things and how many of your needs they could meet, i.e emailing between sessions.

Good luck with however you choose to proceed.