Some of the comments here seem unfairly judgemental - the OP didn't ask anyone to think badly of her husband, she is asking for advice about exiting a relationship which she has clearly put a lot of work into saving - and which still makes her very unhappy. Of course her husband will have his side, but this isn't a detective game to work out who is 'wrong' - she is the one posting - so she should be given respect.
Having an 'exit' affair is very , very common - people who are unhappy will find themselves attracted to others, it's human nature that breakdowns in a relationship often segue into an affair - which becomes the thing that finally gives one partner the impetus to leave.
Yes it would be better if people could leave before reaching that point - but we are all only human - and where chlidren are involved, people often stay well beyond what they want to suffer - until they find they simply can't make it work anymore.
This isn't the 1950's - OP if you aren't happy you should leave, having made every reasonable attempt to work things out through counselling. The OP is entitled to mention her husbands anger and things about the relationship that make her unhappy - it's not about 'justification' - quite obviously anyone who is unhappy and thinking of divorce will have a list of things that make them unhappy - people don't have to believe or disbelieve those things, you can't KNOW from the internet - you just have to accept that is part of the OPs world view.
Yes, considering leaving children could seem unkind - I think it could also be a sign of low self confidence - OP, please don't walk out on the children without speaking to a lawyer, you might then struggle to get the access you want.
I don't think you have to leave a year of no contact with the OM! - seriously, you are a grown up, as long as you leave space to grow as a person and work out what YOU want, when you leave - then you can make a reasonable decision.
If you want to leave, don't move in with the next guy, spend the first few months sorting out childcare/ access/ breaking up as amicable as possible.
Do not stay because of the children - that's never an outcome that makes any child happy - whatever your partner may say in his hurt and anger.
have respect for him and the children - if you keep seeing the new man, do so at a distance, while you get to know him properly.
Its your life, as long as you do so with respect for others, you need to live the life you want. Being unhappy never helps anyone else.