A few little tips and tricks I've picked up:
There's more to life than chasing Balance cards! Got a big multi mob pull coming up and you've just pulled a Bole? Consider throwing it on the tank to allow yourself a few more Gravity casts. Getting constant spears? Whilst it's the black sheep of cards, it's a dps gain for bards and worth throwing their way if you're at a loss. BLMs and Warriors still appreciate Arrow cards it seems. (Not so sure about arrow on Mnk/Nin tho, my static melees would be out of TP too quick).
If you're not parsing, but want to get the most out of your balances, take a note of your party DPS' HP totals and watch their hate generation vs each other, it's a quick way to figure out who's got the most gear and is doing the most DPS and thus who will get the biggest benefit from a Balance.
On the subject of cards, if you are in the habit of queuing cleric stance (to safely drop it without the risk of accidentally reapplying it) after an cast, be a little cautious if you draw a card at the same time, that can sometimes cancel the cleric stance queue if you are far ahead enough with rather obvious consequences.
Don't forget about collective unconsciousness! It's totally free, has a damage reduction element to it and is very strong even if you are only getting the tank offering up more potency than even Aspected Benefic in Diurnal.
Celestial Opposition is oddly versatile and if you have your timing down, can be far more powerful than it's tooltip might suggest. Line things up well and you can extend not only a balance card, but your own Lumi Aether (shroud) by ~35%, regens on the tank and stun a mob pack allowing for a couple more gravity casts. Another good example would be a raid pull timer dropping just before my Draw recast is up, I aoe my saved balance only to pull another almost right after, popping CO not only extends the balance that's up, but also preserves the second balance card long enough for my Spread to be ready before it goes to waste.
If you intend to seriously raid on AST down the line, practice is key, being able to subconsciously juggle your cards is fundamental to performing on point when it comes to mechanics and healing.
Hope these help! <3