Adie is cute, but it could become confused with Addie.I can't think of anything other than what people already suggested for full names.Can I suggest Eda "Edie"? I've been in love with it for a while and have been bursting to suggest it.
I know a man called Adrian who uses this as his nn, so I can't really see it on a girl. But if I were to use it on a girl I'd use it as a nn for Adrienne, not AdelaideS xx