Hi there,
I was in Sydney for a year (now Adelaide). Mosman and Balmain are both very nice, I don't know Drummoyne.
Balmain is full of nice little shops and cafes, and you also have a easy commute to the city (if you need this) by bus or ferry. We have friends who live there and it's lovely, really family friendly fairly upmarket area. Not as posh as Clifton, but in some ways has a similar vibe.
Mosman is considered to be posher, but the main road of shops is a lot busier. Houses are also probably a bit more expensive. Also relatively ok transport.
If you are looking at the North Shore (which is Mosman) I would also consider Neutral Bay and Balmoral. North Shore in general is an expensive part of Sydney though, although nice.
Other good family areas - Rushcutters Bay, Rose Bay, Double Bay are all nice, although the public transport isn't as good.
If you don't need to go to the city, or don't mind a longer commute, have a look at the Northern Beaches - Narabeen, Collaroy etc are cheaper but still nice, with very nice beaches.
Good luck with it, it's a great city to live in.