Do you really think you are dependent in an unhealthy way?

Just because you are emailing??
I think if you are benefitting from the emailing and contact in between sessions, then you are shooting yourself in the foot if you deliberately stop just to prove to yourself that you don't actually need him. You know you don't
need him to survive in the world - but you want to do better than just survive, you want to be happy, so if the emailing is helping your therapeutic process, then don't deny yourself? You trust your therapist, so trust that he too would help spot if he thought you were becoming unhealthily dependent.
I think when we're used to fending for ourselves (in whatever sphere...practically or emotionally or in all ways or in just some) it can be exceedingly hard to recalibrate the new feelings of depending on somebody. It feels wrong and dangerous, but that doesn't always mean it's true.
I know for me, relying on my therapist a lot has actually made a noticeable change to how I am with friends. I am less afraid of being vulnerable with my friends now (still intensely hard and uncomfortable at times, I'm still very much a novice..) and I actually allow my friends to be there for me. Without feeling weak or terrified that something bad will happen because I've let them see how sad or scared I am at those particular times. My friends have said that they
want me to let them in, they
want to help me. On a cognitive level I always would have known this to be true, but I had to learn to depend on my therapist before I could take that leap of faith and extend it to my friendships.