I'm glad you went on the date, for a variety of reasons. Clearly it's not going too badly as you're still with him. He knows the score, so it's all good.
Do Not Have Sex With Him.
It will mess with your head too much. If, in the cold light of day, you decide you want to tomorrow or next week, then go for it, but not tonight. Do yourself a favour here.
Re DH. Do not go on a 'day out' with DS. This is about you two as a couple. It's about your feelings for each other. You HAVE to want each other, it's NOT enough to just want to parent DS together, it's truly not. You need to think about yourselves & each other and see if it's what you both still want.
Be careful though my love, the stage you are at, I'd have probably forgiven my Ex (who cheated on me) if he'd moved her in with us. I just wanted him, us. There are always 'mitigating circumstances'. At the time they seem forgivable and you just want things to 'go back how they were'. Sadly, experience & MN have proven time & time again that that NEVER happens. Occasionally a couple can get past it and be genuinely happy together, but that comes from honestly, and a deep loving and commitment, especially from the one that cheated. They have to accept 100% responsibility. It does NOT matter what led up to it, what you both could have done differently beforehand. HE chose to fuck her instead of talking to you. 100% responsibility for cheating. Nothing less. No matter how much you want to, you cannot brush it under the carpet. It takes a LONG time. In 'real life' I know a few couples who stayed together, but I know they aren't very happy and wouldn't make the same choice again (but feel they can't leave now - whole other story!). Genuinely being able to forgive and rebuild a marriage takes a lot. I tried, I really fucking tried. So much more than I would have, had MN been around in those days! But it tore me apart and he wasn't willing to do what it took really (he very quickly wanted to 'put it to bed'. I lost my shit both over the concept and the wording if that little gem). Eventually we split up.
So really think about whether you can genuinely forgive him or not before you put yourself through the very, very, difficult process of trying to forgive him whilst living every single day thinkng...
(Warning this might be a bit brutal, but you will think about these things. A lot). He chose her. He chose to fuck her. He chose to talk to her. He chose to kiss her. He chose to take her clothes off and make love to her. He chose to talk to her about you and it won't have been complimentary. He chose to lie to you.
I discovered it's totally possible to love someone with your whole being, to want to be with them more than anything, but not be able to live with not knowing when they'd next let you down. It's not the 'sex with someone else' that kills you, it's the not knowing they have your back. Always. Knowing they chose to do something that could destroy you.
Given the things you have said to him, I think the ball IS in your court to say that you weren't ready to talk before, but that you are now. That, you said things in anger that you didn't really mean & you would like to talk to him.
Arrange for someone to have DS & get DH to come through the house. You need privacy for these conversations, not restaurants etc.
Take care 💐