Remorsefully is how you do something when you feel that deep sorry for a real wrong: you speak remorsefully, you look remorsefully at the person you hurt. It is remorseful + -ly, and it is a little formal, but clear. Though actually, it tells us about manner, not about whether the sorry is accepted — you can apologise remorsefully and still not be forgiven. In easy English: in a way that shows you feel very bad about what you did. Put it next to a verb, not in place of I am…