I think you are still reeling from the shock of the necklace discovery. And its taking a while to process the extent of it and you are wisely starting to think about your future options.
You were trying to piece things together for the last five months, but this has rocked you to the core again. I'm so sorry you are going through such a horrible experience. I don't think there's any one way to deal with it. You have to do what seems right to you.
Theres a couple of things which really stand out.
1) "he's done the ‘right’ thing by demonstrating how committed he is to his family,"
He's demonstrating it ... to who.. to you? It sounds like its a bit for show for everyone else doesn't it? He can't accept that he's not the Good Guy. He's lied to you repeatedly, what makes this seem more believable.
2) "He was annoyed he had to move out in the beginning ...but he didn’t actually say this to me" Of course not.. its a play for sympathy, to make sure no one is too sympathetic to you, when he is the real victim.
3) " He says things like ‘It’s in the past’ and reassures me he won’t leave" He is closing down the conversation, and gaslight you. The reasoning is if its in the past what have you got to keep going on about it is the subtext
4) "He doesn’t seem to understand why it would make a difference to me how he felt for her because he is staying and is now ‘committed’.
Again. He would have to be a simpleton not to have a grain of understanding about how hurt you are or simply he doesn't care because he is the self sacrificing and most important person. He pretends not to understand because its a device to close down the subject. He simply doesn't want to hear about your feelings at all. His feelings and sacrifice are the only important ones.
5) "He can’t see that not being absolutely honest (is)storing trouble for the future... it’s all still based on lies if he isn’t being honest now ... he thinks he’s told me everything he needed to)
Of course he does think that! You've clearly spotted, amid all the flannel he packages things up in, pretending to be discussing and reasonable, whilst still trying to shut you down. that its all still based on lies.
OP I may be completely over thinking this but I think this move back and the trying again attempt, the playing the good guy in front of family is all an act, as you've spotted, he appears not to have any thought for the future as the DC get older. He was annoyed to have left the house, because he's realised how much he stands to lose in a divorce, so he's been advised to live at home again.
Why do I think this? because of the way he has hidden his finances from you for all these years. He is secretive and he's been running this show all the time. I also think that the necklace issue could also be explained by him trying to hide assets, perhaps they were in the jewellry shop and he said, may as well buy this before all my cash goes in the divorce. You've been more or less the full time parent whilst he built his career, but I bet you'll find him asking for 50/50 custody, as a bargaining tool.
I really hope I am wrong but I think you should proceed with extreme caution and make sure you get the legal advice to protect yourself and your children as much as possible. He's more interested in living the high life than ensuring your DC's future. You've seen it. Believe it. You have been the one who has made sacrifices, not him. In the meantime, think about the good things that you can do for yourself and DC in the future. Be very kind to yourself and don't let yourself be browbeaten. You and your DC will get through this.