The way that it worked for me with one therapist is that I emailed her, she didn't respond, but would print out the emails to discuss at our next session. That was ideal for me because for years I had such a hard time talking to therapists. I'd get in there and have no idea what to say, but earlier in the week I'd be in all kinds of distress. So I could write down thoughts while I had them and get them out and not forget them before the session. It worked well for her too because she didn't have to try to drag stuff out of me.

I did kind of save up my emails and send them all as one, a day or so before my session. I'd write them as I thought them, but just send it once.

I really think that it is the therapist's responsibility to tell the client if email is becoming a problem, but I can see that you would worry about it, and would probably feel better if you could talk to her and come up with some kind of schedule or plan.