Are you sure she had definitely been crying? Not just had a coughing/sneezing fit or something that made her look puffy eyed and teary?
if she had been crying, it could be for any number of reasons. Crying in her last client’s session. Letting some tears/feelings out from a previous client in between scheduled clients - so she was puffy eyed but not actually crying when she had her session with you. She may have just received bad news about something personal. She may have been in physical pain and cried. Who knows?!
it doesn’t sound like she was still crying/struggling not to cry with you. So, I guess she thought she could manage herself ok for your session. Maybe she didn’t realise how ‘post-tears’ she still looked.
My therapist has got teary a few times in my sessions over the years. Not actually crying, just tears welling a little. When I mentioned it, she reassured me she was fine but that she was very moved by what I was saying. And I felt fine with that. But others may not.
Recently, she did message an hour before our session to ask if we could reschedule to the next day as she had just received some difficult news and needed a little time to digest it. We went ahead the next day and she was fine.
So, if I were in your situation, I think I’d be ok with her looking like she’d been crying previously as long as she was self managing in my session. The thing that would really rattle me in your scenario is her being distracted and engaging with something/someone else on-screen. It’s her job to be completely present for you, to give all of her attention to you. Looking at something popping up on her screen and typing in response would be a total no go for me. If that’s not the norm for her and it was related to whatever the thing was that she was crying about, then that’s a major lack of self-management.
If you like her as a therapist and want to continue working with her, I would probably drop her an email about it. Saying something like she seemed upset last session and that she was also distracted by engaging with things popping up on her screen. And feeling this lack of presence has really impacted you (maybe say how) and that you would like to talk more about this with her at the start of next session?
Therapists are human, of course. But typing away to do with something on screen during a client session) assuming she wasn’t typing notes about what you were saying??) is unacceptable for me, regardless of what may be going on for her. If it was something that needed her urgent follow up attention, I think she should have rescheduled you, even if that came very last minute for you.