I like
Olivia a lot; probably not enough to use because of its popularity though, like you, where I live there aren't so many. And it's a family name, used on a relative back before the First World War when it must have been very exotic. Too exotic for the relative, who went down-market with a thud and called herself
Olive, with nn
Ollie.
I wouldn't use
Liv or
Livia as full names:
Liv because I always assume it's Scandinavian and
Livia because of one of the toughest and most sinister characters in all of ancient history.
Livvy is cute, but apart from Olivia/Olive, all the Olivias I've known have used the full form.