Elizabeth is beautiful, but when used as a fn it generates nns like Liz and Liza which really do not appeal. I used it as a mn for my daughter, and that's been fine.
I like Betsy for its vintage vibe and because it isn't Liz. I knew a Betsy at school, and liked her name much better than her.
Elizabeth Jane is lovely; very Thomas Hardy but there's nothing bad about that. (E)LIZ(abeth) and (Lou)ISE would bug me with their echoing sounds.