Gosh your DH is really trotting out the standard party line. From the things you've said, he is acting in a truly, predictable way.
Angry when you first questioned him
Distant and runs away when caught out
Blaming you and the fact that he has been unhappy for ages
Shocked, I am so sorry that you are going through this. There is nothing anybody can say to make you feel better - try to keep on talking here and please, please lean on your RL friends and get some solid legal advice ASAP.
FWIW I think his immediate reaction to being caught is hugely damning of him (both as a husband and father). You have to protect yourself from further hurt, so if I were you, I'd take control and pack the remainder of his things and send them to him. Even it's not forever, at least until you have had the time to clearly think about what you need. I know this will be incredibly hard but he will mess you around if you don't take a firm stand. My other piece of advice and sadly it's based on experience is not to let him see you cry or get upset. Don't text him, don't contact him, other than to do with contact for the children. Even if you are scream and cry when he's not looking, don't do it in front of him. There is nothing more empowering for men like this than seeing the effect of their actions.
When all else fails, run a hot bath, cry your eyes out until there are no more tears and then put one foot in front of the other. These first few days will seem like years, but before you know it, weeks and months will have passed and the initial horror will subside.
Take strength from your beautiful children.