It sounds like she's trying to help you find a place in your relationship with her that is realistic and that models what happens in most "real" relationships.
What I mean by that is we all have times we feel like we need support. Sometimes the people in our lives are right there, immediately availabje and ready to step in and help. At others we really need to say we need help and communicate what it is we need and how quickly day by day. There are also times when the person we'd like to help just isn't available when we need them or isn't the right person.
None of those things mean your need is wrong - the nature or relationships are that we can express a need and the other can decide whether they can help or not, based on their capacity and availability.
It sounds like your T is being very honest about what she can offer. She can't promise to be available when you would like her to be and that might change day to day - which is what happens with most people, we rarely make rules about how quickly friends or family will respond to something, but if we need them immediately it's on us to communicate that. For example, I know a few people who check their emails every few days who wouldn't see something unless I let them know I sent it, so if it's important to me I know I need to flag it to them.
I know it's hard but it sounds like she's trying to help you identify when you have a need and to express that by telling her and telling her what you need. I rarely make contact with my T between sessions but when I do I'll tell her what I need. Usually I'm just letting her know something that's happened that's important to me and that I don't want to surprise her with (e.g. When my mum died recently I let her know so she had time to think about before session because we had just started a particular piece of work and I needed to focus on her death instead). At other times I'll tell her I'm just thinking out loud and can she bring it to next session. She'll never send a full response - just a quick acknowledgement and a reminder of when our next session is. That's fine, it's her boundary and I'd rather she took care of herself and didn't burn out.
Not reading your emails at all isn't ok and may suggests she's too busy generally. Not too busy for you specifically, just too busy. If she has said to text her and, if you need an immediate response tell her, that's what I'd do if I needed a quick reply. It might be worth exploring what she means by urgent so you have a clearer picture. It's ok to find all this hard - that's why it's called "work", she's trying to help you change your patterns in relationships and that's very scary and sore.
I also don't think you need to maintain a triggered state to make what you've sent her still be relevant. I've learned that part of me writing when triggered was that it helped me see what was going on in my thought process and get it down on paper for a record - because in my not triggered state I felt completely differently and almost couldn't remember what the fuss was about. Writing it and sending it meant my T kept it safe, I'd feel less activated which meant daily life felt easier, and there was a lot of benefit in going back and looking at what my thoughts and feelings were when triggered -v- when I was doing ok. I still struggle at times in looking at where my mind goes to when I'm upset or angry about something but it's part of developing trust in her to know that I can bring all of those parts of me to therapy and know that I'm accepted there.
