Accidentally posted this on your first thread rather than here, sorry! I commented on your first thread but I was just reading one of your subsequent messages and wanted to add another bit of advice. You need to make it clear to him (and yourself) that if he holds any hope of staying with you then drip-feeding you bits of the truth is the quickest way to damage any chance of that. He needs to own up to everything, up front, or forget about being with you. Not in huge details but at least the full extent of the relationship. You will probably never fully trust him again but if he does his best to show you he is genuinely trying to right himself, that is what you can try to trust. That's if you do still want to be with him yourself. It is possible to go on and have a very happy relationship after cheating, but only if the person responsible genuinely wants to and does change their general behaviour and works to understand why they did it so they can avoid doing it again. If he's not willing to do that, don't waste your time and energy.
I'm so, so sorry once again, especially with your children being so young. I was glad to read you are being careful not to use them against him — that is very much the right thing and I hope he recognises in that what actual maturity looks like. What he has done is totally unfair, selfish and immature of him. One more thing: I understand why you would not want to broadcast this to your wider social circle, especially if you hope to stay together. But don't protect him from the disapproval and anger of your closer friends and family, including his friends. He has not just hurt you, he has hurt your network of people and he should be held to account by them too. Don't let him get away with hiding behind your hesitation to tell others. You deserve their support, he deserves their disapproval!
PS If it is bugging you not knowing where he's gone, can you see on your phone or has he stopped sharing location? If the latter, I'd remind him that if he hopes to get back together then he needs to start sharing his location with you at all times as part of proving he's going to set things right. That means from now, not from when you let him move back in (if you do), otherwise how are you to know he wasn't with her in between?