First suggestion, don't take brawl competitively
...but you already play melee that way so. Next learn how to do nothing, in brawl and with ZSS too in many matches EVERYTHING we do is unsafe yes running away over B is normally safe but you run out of stage that way, and you really need that stage, don't waste it to throw out 2-3 over Bs.
With ZSS I would tell you to learn how to play for 1 week only using, d-tilt/u-air/b-air/f-air/down B...and up B for recovery.
Once you get good at using those then move on to using u-tilt/dash attack/over B/d-smash/up B aggressively then finally n-air/jab/f-tilt/d-air/f-smash/u-smash/neutral B.
It might seem restrictive at first but that is the point I see WAY too many ZSS's that rely on u-tilt/dash attack/over B/d-smash/up B offensively way to much
While n-air/jab/f-tilt/f-smash/d-air/u-smash/neutral B are useable moves, they are a bit more of niche moves. I sometimes see people using these moves way to much too normally the culprits are neutral B and u-smash.
As far as movement goes we are great in the air, and we are good at screwing up peoples ledge games, so punishing someone to a ledge is a very viable strategy, where they either have to jump guess on a punish or run off the edge.
Try to get them to the edge not so much by forcing them there but by being tricky with positioning, like I said we really don't have safe pressure.
Don't forget we have a great f-roll, and because you can just dash back then roll a good backroll as well, you can use this sparsely to trip up your opponent.
It's the closest thing we have to a ground kill-move.
I use it. :D
I use it a lot, it's a green ZSS thing lol
Plus it ***** airdodges, so when they're hit they don't expect it and DI horribly.
GASP are people finnally non-ironically agreeing with me that while f-smash is bad it has a purpose.
Basically yeah it can kill if its fresh(it always is), and its meaty(lots of active frames), so its really hard to dodge. But yeah its slow 20 frame startup, and if you miss it your ****ed.