Hello Crying. I'm so sorry to hear you are going through this.
I'm by no means an expert but it seems to me that the sooner you can separate from him the better. I realise there are financial implications but you will be much better off from an emotional, mental and psychological viewpoint than being in the situation you are in now. Because it's like rubbing salt in your wounds. Every day you are feeling more heartbreak and confusion because your H "feels better" and is therefore behaving better, giving glimpses of your happier past and the way it used to be. No wonder you are confused.
I know you're not looking to rekindle the relationship but you really do need a clean break so that you can get the healing process under way - this isn't going to happen while you are still living under the same roof.
Why not look into what benefits you may be entitled to as a single parent? He will also have to pay maintenance for your DCs. The situation might not be as bad as you think and it may just be possible to make the break financially. Do you have close family or friends nearby who can offer support?
Can you ask him to move out? He should be the one to leave as he is the one who no longer wants the marriage and you will be the one who cares for the DCs. If you have a mortgage, you would be allowed to stay in the house till the DCs are 18, as I understand it. I'm afraid I don't know what happens if you are renting. There are loads of women on here with far more knowledge and experience than me - try posting on the Divorce and Separation board for help.
I don't want to add to your distress and I can't say for certain whether there is an OW, as there may not be, but in my experience men very rarely end a marriage when there is no one else involved.
It takes a long time to emotionally process and properly get over a break-up, which includes figuring out what you want, and you can't get this process under way while he is still in the house being Mr Nicey-Nice.
I know how hard it is (I really do!) but please try to detach your emotions in relation to his changed behaviour. How he thinks, acts and feels no longer matters and doesn't change anything - he is the one calling time on the marriage - what matters now is self-preservation. He should do the decent thing and move out.
I've been involved in many break-ups throughout my life, and whether I was the one who ended it or not, I found I was always better off getting away from the other person and limiting contact with them as far as possible.
When I was the one with doubts, distance gave me clarity. When I was the one devastated, distance gave me peace, dignity and, over time, strength. Distance is your friend. He is not. Please try to get him out of the house my love. The DCs are young enough to adapt. Thinking of you and sending you big hugs
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(apologies for the long post - I have no idea how I manage to write so much!)