It looks like you’ve had loads of good advice on here, OP.
As someone in a really similar situation (but bigger debt 😬) I just want to echo the advice to insist on financial transparency, and financial safeguarding for you and the kids.
I’d be really wary of believing him about all the details, they take a while to come out (and possibly never all do…) and “I’ve stopped” isn’t always the truth.
You've had a tremendous shock, and it’s perfectly ok to not decide whether to stay in the relationship until you’ve taken some time to recover from the initial blow.
He absolutely needs to engage with support on an ongoing basis. And I’d really encourage you to tell other people. It’s a massive thing to carry on your own and secrecy allows him less accountability. If he’s not willing to tell people, this might be a red flag. Relapses do happen. And addicts are great at lying.
There’s support available for you too - call Gamcare’s helpline and ask about their women’s pathway for affected others, for example. But, as I’ve said to others on here, make sure any support you get is about YOUR interests. There’s plenty of advice out there on “how to support the gambler in your life”. Gamblers do need support, as it’s shitty being an addict. But the people whose lives they wreck without knowledge or consent need support for themselves too.
As others have said, the relationship will never be the same again and the trust thing is very tough - you’ll have to work out co-parenting and cooperation for the kids. Anything beyond that is absolutely your call and not to be committed to in a hurry.
It’s been several months here and I’m still in the process of all this, so sending solidarity and strength. It’s really shit. Best of luck.