I'm sure you've read enough threads in this part of the board to know that most abusive men do this. There is a pattern to the behaviour. They will do whatever they do, then be sorry, honeymoon, then abuse again. Abusive men will say whatever they think needs to be said to get the pressure off, to placate you, to get you back into that nice submissive state which makes his life easy but completely messes yours up because he keeps telling you that black is white and you have to try to believe him because you're so confused that you think he might actually be right.
I really hope he is sincere, I really, really do. I don't think he is.
So let's say, for argument, that he genuinely means it; that he really will try to change himself. It is not easy and will take some time. In the meantime he will almost certainly behave as he always has most of the time, because that's his default setting and that's what makes him comfortable. So you will be on the receiving end of it, as usual.
This is not going to help either of you, or your children. It won't help him because there is no incentive for him to make the real effort to change, the effort to understand that you deserve, and have the right to expect, respect from him. It won't help you because you will still suffer from his abuse.
The only way to sort this out is for you to live apart from each other for a while. This doesn't have to be permanent unless you want it to be, but you may not be able to make that monumental decision at the moment. What you need is space. You need to be safe, somewhere where he can't dominate and control you or the children. Really, that means he moves out, but you may need a refuge or some such - how likely is it that he will agree to move out until he has his behaviour under control, adjusts his thinking and actions so that he is able to make you feel cherished and protected?
You are going to need quite a few months away from him in order to get your head together, to remember who you are, to remember that when you see black you can call it black, and not struggle to work out whether it's white.
It may be that WA will be able to give you and your lovely children a place immediately - take it. Please. You may have to wait for that a little while, or you may just go and rent somewhere using housing benefit (to which you will be entitled - WA help you sort out benefits etc).
May I also remind you that the most dangerous time for an abused person is the time when they are preparing to leave. I am not saying this to scare you, but you need to be aware. This is when the abuser brings out the big guns - the charm, the presents, the kindness, all the things with they hooked you with in the first place. If they don't work then the reversion to being the bad guy is swift and strong. Men who have not actually hit their partner before may well do it at this time. Don't mention WA to him, and if WA tell you you need to go now then go.
Beautiful, you have tried so hard and you do not deserve to be in this situation. God loves us, and he hasn't put us here so he can watch us being miserable or being destroyed by life; he wants your life to be filled with joy so that you can love him with joy in your heart. You deserve someone who will genuinely love you, whose life work will be to protect you and your children, to make you happy, where just your smile will make his heart sing.