I used to have this problem with T a few years ago when she said I could email her. I had such huge anxiety waiting for her reply, checking my email 50 times a day...one session she told me to email if I needed to and she would definitely respond. She didn't. I arrived in tears for the next session as I was so gutted. Thankfully she apologised and recognised what a big deal it was for me. She then made a promise that she would always reply, even if just to say that she didn't have the time yet but would reply the next day/over the weekend etc. She has also told me that if I haven't heard from her within 2 days of emailing to resend it and check that she got it. She has kept her word and I am no longer anxious.

So I do think you need to discuss with him how the uncertainty around his replies makes you feel and maybe you can then readjust the expectations together. Sometimes just knowing someone has read the email can be helpful, even if it's just a one line response to let you know that. I hope you manage to figure it out.