I read your post and couldn't find a reason to feel ashamed or embarrassed.

You contacted your T as you were having a hard time. Sometimes we contact our Ts a lot. It happens. You will not be the first or last client to do it, guaranteed.

Your T self-disclosed about his mother-in-law. Why do you think he did that? He is a T. His personal problems are not your concern. So why would he tell you?

Answer: so you understood why he hasn't been in touch, and didn't feel rejected or abandoned as a result. He didn't leave you to wonder why he hasn't responded, then tell you ages after.

He didn't tell you to make you feel bad. He told you so you knew why he wasn't responding.

Those are not the actions of someone who thinks badly of you. He must care a lot about you. You have done the right thing by apologising. Now please please forgive yourself.