Blamelessly is how you do something when no blame should fall on you, or how you live when people cannot fairly say you have done anything bad. It is more formal than blameless, and quite rare in everyday talk. You are more likely to read it — she lived blamelessly — than to hear a classmate say it. It does not mean you never drop a cup or forget a name. It means, in the speaker's view, you are not guilty of anything wrong. Though actually, writers sometimes use it with a raised eyebrow, as if perfect innocence is a story people tell about themselves.