You are supposed to feel happy! You deserve to feel happy. This is good for you and your children.
Nobody and I mean nobody gets anything from you being unhappy. You don't need to punish yourself for doing what you know is right.
You are separated and will be for a while, perhaps you will never be a couple again, no way you can answer that right now and absolutely no use worrying about it. If you feel you have already made that decision, perhaps sit with it for a bit and make sure it feels right to you. In the meantime - yes, be happy!