Seeking expert counseling is a sign of strength. A personal relationship with a caring professional is proven clinically effective.
I believe that I can help.
Sometimes marriages fail. Yours was an error of judgment perhaps. You did not know how quickly it would fall apart.
There is no blame for people's inability to get along and build a happy life together.
You have tried to get back with our wife but it seems to be way too late. You could give up the woman you love and do the dutiful thing and stay with your wife but you will resent each other and provide a very unhappy household environment for your children.
It seems to me that no counselling as individuals or a couple will help to do anything other than to establish the best way to separate the family, make sure the kids are supported, and the letter and the spirit of the law are followed.
Hopefully you have found a partner that will keep you happy and be a loving and supportive step-parent to your children.
Hopefully, your wife can be helped with her condition, which may be depression, or even a personality disorder or something else (you said that her behavior was odd).
From your words and manner I hear a decent and caring man in a terrible situation. Yes you cheated, but you must forgive yourself. You owned up to it and have great remorse. None of this can be undone. You cannot move backwards so you must move forwards, pass this unhappy situation in which all of you are currently immersed.
I highly recommend that you and your wife go to a marriage and family therapist. Listen carefully and talk to each other.
Don't accept a reconciliation that you know you cannot live with but listen.
If you decide on divorce, make it a gentle one.
If you can get hold of this book in South Africa (or order from the USA or UK) then I recommend it:
I wish you great success in resolving this with as few tears and pain as possible and I shall keep ALL of you in my prayers
Elliott, MAE, LPCC, NCC, CCMHC