Oh my heart goes out to you solost. I have been EXACTLY where you are now. Your posts ring so true for me.
You want your husband back. The man you thought he was.
You're making excuses for him. Blaming everything else, stress, work, anything to help you come to terms with how he could have done this.
You're allowing him to come and go as he pleases. So he is.
And how dare he check up on you and get angry when you go out with friends!
Now I don't blame you for any of that. I did it all and felt exactly the same! But - now I can see it from the other side I can see what an absolute tw**s these men are!!! 
He is most definitely having his cake and eating it.
It's ok for him to walk out with a few days notice and be with the woman he wants. Knowing full well you will pick up the pieces with the children and he'll know what you're up to 24 hours a day! And when he doesn't, he will call and text until he does!!
As for the wedding ring, my ex did the same, always wore it. Made me think deep down he just couldn't bare the thought of actually losing me and valued the marriage.
Bollocks! I later found out that the minute he left my house he would take it off! And the one time he forgot, the OW went mad at him so much so that he pretended to throw it away to appease her!!!
Please please please, do not try to belittle how hurt you are or how much you want your marriage back and the man you married. But at the same time, take away that respect you had for him and put it back onto yourself. Respect yourself to set some boundaries and tell him that if he wants to be with her, he can bloody well be with her. Full time.
And show him the FULL consequences of that.
I had 4 years of this. I allowed him to come and go, play the good husband for a few days, then the confused person, then he was stressed, then he wanted her, for 4 years!!
Please learn from my mistakes and try everything in your power to take the control back in this situation xx