I'm so sorry you're going through this!
I know exactly what you mean. After I recently moved from another state, I had to leave a t that I had an amazing relationship with. For the first time I truly felt safe opening up, (though sometimes I needed a lot of encouragement from her to do this) and cry, and do all the things I usually don't do because I keep a lot hidden. She did hug a lot, and would hold my hand or even hold me in her lap.
I'm just starting with a new t. I don't know how things will go, we'll see. I know it'll be different, and she might not be as warm as first t.
To me the connection is so so important. I think things like the e-mail thing are controversial and not deal breaking things, but if you in general feel like she's a "robot" and you can't connect with her then how can you ever do the work needed to heal?
Even though it's hard, I would consider looking for someone else if you can. Please keep us updated. If you want PM me and we can talk more on this.