Uh, no he doesn't have the upper hand, financially.
Firstly, conduct is irrelevant in divorce so this behaviour of his won't signify in any settlement, so it doesn't actually matter if you have the evidence or not. Reasons for divorce petitions are not public and he can't actually defend it anyway - all he can do is state he disputes your reasons but consents to the divorce - bam, done with. What will signify in monetary terms is that you've been married over 10 years and that you have made significant career sacrifices to support the family, and are primary carer. The starting point for division of family assets is 50/50, plus child support. If stability for the kids means you get a bit more, that's not exactly unheard of, either. Your kids are young and if at all possible it's preferred that they stay in the family home, too, so he may have to leave and you may get to keep it - all fairly complicated but NOT bad news for you. Don't panic about that.
Please see a solicitor first thing tomorrow - a good one. You know and he knows, and if he wants to be a complete bastard by denying this, attacking you, and telling himself he was justified anyway, then let him. You can ensure your kids are secure financially and you'll be just fine, in time. A solicitor will mean he realises very fast that you mean business and he could lose everything, and he may be rather more penitent when he's had time for the shock to wear off. You don't want anyone to know anyway, so if worst comes to worst he can vent and foam at the mouth while you divorce him for unreasonable behaviour, while you remain dignified and silent. I know who's going to look the one with something they want to hide. He's a fuckwit, you are not, sod the tosser unless he starts to put his big boy pants on and actually talk to you like a human being.
Is there a laptop or PC he uses at home? Some MNers are very technically savvy and there is likely to be evidence on more than just his phone, frankly. I have no clue how to obtain it but I know others here might.