Hugs if wanted. So, this is a case of your T failing to manage her own boundaries, which she is responsible for. If she's going to sit down to watch a movie, then it's her choice to look at her email first. You have no way of knowing, when you send an email, what she's going to be doing. So it's not your fault for sending the email. She should not have told you this, as it implies that it is. It's up to her to decide "OK, I'm going to check email before doing this," and then if she's distracted, that is her choice.

I've run into major conflicts with T's both past and current regarding email (and other outside contact), so this is a particular sore spot for me. With ex-T, I used to send long emails (and if she replied at all, she'd just say a sentence or two and that we'd talk in session). She never complained about them until I mentioned her not responding to one that I thought was more urgent, and she said something, "I do have an outside life, and you're not my only client." Which really hurt me (I had some maternal transference for her, mostly negative).

With my current T, we had a rupture a couple years ago where I emailed him in distress on a Friday night. He generally only replies to emails in the morning, so I wasn't even expecting a reply that night. He responded with something that further upset me, I replied and said that, and he then responded to me with something that upset me even further. What made it all worse is that in the next session, he said he'd felt "trapped" into replying to me that night. It was partly that I was in a bad place, and I think he was concerned for my safety, but he could have also chosen a path other then sending me multiple emails (a brief phone call to evaluate my status, telling me I needed to go to the ER if needed, saying he couldn't help me that evening and to call a crisis line instead, etc.). But he chose not to (we've since worked all that out, but it was very painful at the time).

That's probably more info than you wanted or needed, but I'm trying to say that I get it. The stuff about her kids, I can see how it would make it that much worse. Especially using the word "protect," as though you're some sort of threat. I would talk to her about this. Tell her how you reacted to what she said. You can include that it's coming from a young part of you, mention the childhood wound, the maternal transference. It may be difficult, but I think it's better to tell her and talk it through.

I think maybe it's good that you're changing T's, even though I imagine it will be difficult.