It is very heartbreaking to read this...
Last year, prior to getting pregnant this year, I also caught my husband cheating on me (although there is no proof that he physically cheated and he also denies having sex with third party, but I caught them going out for a couple of months).
I spoke to many friends who went through cheating, with & without divorce and thought about it thoroughly.
Decided to forgive him because he showed remorse and really show me action that he's changing.
Fast forward to 1year after i caught him, i still often have fears and doubts too, but can tell he is trying his best to be transparent and letting me track him (face time anytime) and give me all his passwords. One important thing i did was to make sure his salary goes to our joint account so I can track where his money goes (of cos if he pays anything by cash, i cant do anything). This salary arrangement was made after i caught him, and it's one of the criteria i asked him to promise if he wants to stay together.
In your case, it is quite clear he has physically cheated. I can understand why you are thinking of forgiving him, must be due to the kids. Most comments will tell you to divorce, in fact, I think I will tell u the same if you are my friend, simply because your husband does not seem remorseful, nor is he committed to reassure you, or try to remove your fears and doubts about him (no apology, no change in action, and even change his password to hide from u further).
If he at least did the above changes, still can give a chance, but clearly, he's not working on the marriage. Maybe he assumes u will definitely forgive him for the kids? You may need to give an ultimatum.
I will suggest that you make preparations for the separation as it seems too difficult to trust him again, as it is not easy to just leave like that with the two kids, and especially if you don't have income enough to support the two of them and yourself.
A few things you can do:
1. try to gather as much evidence as you can, always cover your own backside:
e.g. i took photos of all his chatlog with the 3rd party, and conversations he and his friend had, who as mutual friend with third party.
Whatsapp chatlog i emailed to myself and went to his email to delete the "sent".
However, if u dont have his PW now, it is quite hard to do. If I were you, I will engage a PI to do this if got budget.
2. find a new job that can support the two kids and yourself as much as possible.
when fighting for custody, i believe this will come in handy.
3. speak to a divorce lawyer on what are the steps you need to do, documents you need to prepare, etc.
#1 the evidence is so that you can ask for more monthly alimony.
however SG law - no clear photo/video evidence of him "in the act" means no evidence of commiting adultery. this is the hardest to get.
be strong and be brave. jiayou!!!