I've been in a similar place as you before. Not with my therapist, but psychiatrist. We didn't establish any ground rules before though. Email was our primary form of communication. No office phone or text, although he did give me both. He works in a hospital though, primarily doing ECT, so he's almost never in his office, doesn't have a secretary to take messages, and it feels weird texting — too personal, so I try to avoid it.

Long story short: there have been times I've been a little too excessive with the emails (not as frequently as you it sounds, but more than I normally would between sessions). A lot of times he would respond, usually within a reasonable amount of time, even weekends and in the evenings, but sometimes he hasn't, which (I'm not going to lie) has been anxiety-provoking and/or hurtful. I never brought it up to him, probably because it was too embarrassing that him not replying mattered so much on even the most trivial of things, including scheduling follow-up appointments. I just assumed and told myself that maybe he was too busy at the time or he was planning to address it next time we met. Over time, though, I was able to get a better feel for him and realize that he's a busy guy at work and has a wife and two kids outside of it, so sometimes he misses things or sees it and then gets distracted and forgets to go back to it. So, that's made it a little easier to tolerate.

But to reiterate as others have said: asking your therapist to not read your emails is ineffective at best and undermining at worse and could interfere with your (what sounds like generally) positive relationship and takes the control away from you, even though it might not seem like it. Also, as others have said, whatever is going on, some subconscious process is at play and I think it would behoove you to really dissect and talk about it in therapy. Obviously I don't know your life story, but maybe you had a tough childhood and one or both parents were absent or neglectful to your needs and because of this your attachment style in interpersonal relationships took a hit. In other words, maybe subconsciously you treat your therapist as one or both parents and get distressed when your needs aren't met i.e. reply to your emails.

Just my two cents anyway. Sorry for the novel.