You need to see a different therapist or explain that you are being crushed by guilt and ask them to help you. Sure, break up if you want to. Maybe a fresh start is what you need.
Having affairs is not unforgivable. I think you are missing the point.
Something is wrong in your relationship if you are going outside of it for sex, love, outside attention, an affair. That should be your concern. Why did you do it? Not beating yourself up for being a human and making a mistake. Why did you cheat? My guess is you are not in the right relationship that satisfies you or fulfills you very much. It’s not a good match. Maybe he’s nice and a good guy but he isn’t The Guy for You, and this is why you cannot forgive yourself. Let him go, seek a new therapist and try to reflect on your behavior and treat people better next time.