Texting and emailing aren't really the same thing. Some people don't do email at all over their phones, so just because he is texting doesn't mean he's checking and answering email. And even if he's texting, you really don't know who he is texting. It very well may not be clients at all.

You said you've "gotten the impression" and that you cut your session short because you thought he "was looking bored". You interpret no response as a "clear response that I am a burden and an annoyance and a pain to deal with". You assume "he's never been this alone". Lots of assumptions and mind reading going on there.

Does your therapist actually do therapy via email? Have you had that discussion with him? It sounds like probably not since he told you email is okay "within reason". Usually that means feel free to send it, but it isn't his preference. Historically he doesn't give lengthy replies, so he'd probably rather you call or discuss things in session. Many T's work that way. I know mine does. That doesn't make him a bad T, but perhaps it isn't the style of therapist you prefer to work with. If that is the case, you could consider finding a therapist who is more open to technology as a tool for therapy. They're out there.

Talk to him about how this frustration fuels your abandonment issues and see if there is a way to work with this. He may not offer anything more than he already does; that is his professional choice. But you never know; he might come up with some options for you.