I agree, it's time for you to work hard on regaining your H's trust in you. My H expects me to trust him again without really giving me anything to go on.
If you have to show him your email, then so be it, if he needs details about your affair, that, too. He needs your reassurance that it's for real with you. Let the BF have notice to move out, and let your H know about it, and don't ever have contact with that person again. Both of you got the M to this state, be grateful he hasn't moved on and wants to work on it, so give it your all.

L