There’s a difference between lying and keeping your personal life private. But I don’t think that’s the real question here. The reason you are feeling so anxious about sharing your relationship with your family is that you know in your heart that this situation isn’t going to end up anything but bad. You are with a guy who is cheating! You know that there is something wrong with a guy who will cheat rather than deal with his marriage honorably. That means seeking some counseling or separating cleanly, not sliding into a relationship on the sly with a younger woman. Further, you are violating your own values. You say you aren’t the kind of person to hurt someone else but you are doing exactly that.
An important part of being an adult is being able to make good choices in spite of our feelings. Little kids go after what they want because they feel like it or because it feels good. Adults are able to delay gratification in order to meet longer-term goals.
If this guy truly loves you, he’ll resolve things with his wife and do some personal work to learn from a breakup before he asks you to move in with him. If you love yourself, you will take a huge step back. Are you really willing to betray your own values in order to be with a man who isn’t man enough to deal with things in a mature way? Think about it.
I wish you well.