Sounds to me like you are having a really hard time accepting who you are and what you have done, and that yes - this REALLY IS your reality now. For me, coming to terms with the fact that there really is no running from this, and no matter what I will always have done it, has been central to forging my ability to push through the pain, the shame and the depression.
Your feelings are normal. What you gave done has caused immeasurable heartache and has the potential to in the future. I know I would feel like my world was ending if I ever had to reveal to my son that I saw another woman while his mum was carrying him. We SHOULD feel agonising pain over this. Hopefully its what will transform us into people that will never let it happen again.
Once you find the clarity on the otherside of this fog, to look yourself in the mirror and accept - yes, I cheated, yes, I betrayed my children, yes, I have epitomised selfishness and yes I knew it all was wrong.... THEN you can make a resolute choice to be the man you should have been all along. One who owns his mistakes, but pushes forward in a spirit of integrity, love and truth hence forth. Even if they hate you, even if they leave.
Your chances of keeping your family just took a massive hit. Just like mine did. But you can ONLY increase them by accepting and living the above.
All IMO obviously. I hope you can come through. And I hope your family comes with you.