Yes, definitely don't start digging around the galley on your own. Even if something is laid out, its not good to grab n go, especially with ice (non-canned/sealed item).

If you're in business or first, I don't think you'd have a problem letting the FA know in advance and they'll try their best to keep your glass filled. However if you're in Y, I'd suggest buying your drinks before you board if you want to make sure you stay hydrated. Simple math - the ratio of FA's per pax in Y is alot higher so they may not be able to consistently get to you in a timely manner.