I disagree with those that say you need boundaries about how often you can email. I think you just need a conversation about it, where she reassures you whether it's actually OK. You can also talk about what you expect of her (don't read it, read it and don't respond, read it and respond 'thanks, see you next time', read it and reply in-depth?) and what she offers.

A while ago.. in the Christmas holidays I think.. I was very agitated and stuff, also couldn't stop talking, and my pdoc mentioned he'd also noticed that by my emailing frequency. At the time I thougt that wasn't related (although the length of my emails was) because the additional emails were about appointments I had had (some medical issues) and I often write about them to give a short summary. Because often I bring up/mention these issues to my pdoc first, or he's the one that advises me to go to my GP, or whoever I saw has advice or medication I want to discuss with him before taking.
Now that I'm less agitated, I do realize it had some influence. Giving a recap of everything said/advised, and also things that weren't so important or need-for-detailed that it would have been an issue if I had shown up at the next appointment having forgotten half the other guy said.
But he never said I shouldn't email. Not sure he would, anyway. If I were to email three times a day, I think he would - depending on the issue - either see it as a sign of how bad my symptoms are (and work on them), or advise to bring the issues I'm mailing about up with my therapist.

As for my T, I'm allowed to mail as often as I like. She used to be part of a team in the big T organization she works for that attempts to keep people who 'need' inpatient therapy, home. They try to reduce the frequency and length of inpatient hospitalizations/residential treatment by providing 24/7 support (phone, what's app, sms, email, as well as increasing sessions as needed) when needed.
She isn't part of that team anymore, now just handles complex people (it actually says so in her e-mail signature. "Complex problems and DBT". She also does DBT), but said 'nothing will change for you'. (I think some things HAVE changed - she's less available - but not the emailing part.)

Again.. I don't think emailing a lot is necessarily a problem. Your anxiety about it is is, and if you feel it's 'maladaptive' or your emailing is standing between you and autonomy, then you might want to do something about it. If not, just talk it out.