Ease his mind about my “cheating”

I never cheated on my SO but we did have some issues about lying and I was covering up some advances my sons Dad threw at me SEVERAL times. Also back in April, while my ex was going through his break up, I was his crying shoulder and my SO found out. it wasn’t emotional cheating at all bcus there was boundaries but when my bf found out and put everything together, he says he thinks we may have Bcus he was too comfortable in trusting me with his issues. And also one night he called me really late at night for a ride home after work, knowing I live with my partner. We want to work I.t out so i dressed up about deleted messages and all the advances. I fessed up about being his friend through tough times, and now my bf thinks there could be more. He wants me to somehow ease his mind and really prove that I didn’t cheat on him but I have no idea how to. We are about to have a baby and I feel like my world is falling apart completely without him. Sucks but that’s how I feel at 9 months pregnant and lonely and having moved back to my moms. If you were in this situation l, how would you prove or in essence ease his mind about you not cheating. What do you say or show him? I even thought about texting my ask casually but I realize that looks and sounds so stupid, someone I barely speak to as I.t is, like hey when’s the last time we slept together? What should I do?