I would never use Anise because it's pronounced a lot like 'dill' is pronounced in French, and I don't like dill. It would be like naming my child Liquorice. (Another taste I don't like...) Based purely on sound, Anise is OK though. It might get misheard as Elise sometimes though.
Felicity Anise - flows well IMO
Gwendolen Anise - so-so
Josephine Anise - sounds a little rhyme-y
Melanie Anise - again, so-so
I love the name Molly. (I've never heard "Molly on the Shore" though... I might have to look it up) I am bothered by it's growing popularity too.
Molly Celia - doesn't flow right to me
Molly Clementine - cute
Molly Elspeth - I like
Molly Helen - nms, but nice, especially as it's honouring
Mally Karen - I happen to strongly dislike the name Karen
Molly Susannah - nice
Ada Molly - I don't like these two names together
Enid Molly - a bit old-fashioned, but not bad
Isadora Molly - very pretty
Susannah Molly - also nice
I like:
Molly Delilah
Molly Theodora
Cordelia Molly
Oriana Molly
I like most of them with the first/middle names switched too.