You say you wonder if you are doing the right thing, but the title of your post is “How do I find the strength…?” That says it all. You know you are doing the right thing.
The right thing is to give your children a foundation of love and commitment, and you can’t give that to them if you stay in a marriage where there is no love, or where you have emotionally “checked out” just so you can keep the proprieties of Mr. & Mrs.
I left my first relationship, and it is so much easier than you could ever imagine. It is simply a series of steps out the front door, and into the car. I know you think about it… every time you drive to the store with the kids, just continuing on that drive to another place. It really is that simple. There’s no more emotional trauma involved than there would be if you stayed in that home and tried to create a new life with the man who had an affair… and who has continued to keep that woman in his life, on some level.
Leaving seems so hard until you actually do it. And then you realize, it’s just like packing for vacation. Get your junk, and go. The rest falls into place. If you don’t have your own furniture, www.freecycle.org. Everything there is free, the only stipulation is you usually have to pick it up yourself. I furnished my entire post-separation apartment - from couches to silverware - on this site. You’ll make do with mis-matched curtains until you can get it on your own, but the first step is walking out the door.
And it’s the hardest step you’ll ever take… but in retrospect, it will be the easiest thing you’ve ever done, I promise.