Seriously, dude - you took in a woman with three kids, offered to raise them, and she spends her time sexting and cheating on you - when she already KNOWS how that hurts, having been a victim of infidelity herself?
WHY, on EARTH, would you want to take her back?
Look - people who are married, have kids togehter - I"m all for doing everything possible to salvage the relationship. But you are NOT married, these are not YOUR kids (although I'm sure you have feelings for them) - and she's ALREADY SHOWING YOU that she has poor moral character. Why on earth would you want to get further involved? This is not a person you can count on to be your support if you lose your job or get cancer or some other catastrophe of life befalls you.
Now - we all have things we need to work on in ourselves, and sounds like she has identified some things in you that you agree need fixing - go ahead and work on those. But also work on why you feel you deserve so little, that you would want back a woman who treated you so badly? And a woman that is such a bad mom that she would move in with her OM and his parents with her kids? (Seriously bad mothering there.)
You might have a problem with White Knight Syndrome - some guys are addicted to "rescuing" women. The problem with that is, once you rescue them - you're stuck with a woman who can't manage her life.
Or maybe you just don't feel very successful in love, and felt like she was the best you could do?
Whatever it is, THAT is what you need to work on here, my friend. Why you feel these crumbs are enough for you. You need to feel that you DESERVE a loving faithful competent partner who feels that same way about you that you do about her.