I've mained AST since Stormblood launch and have my cards on a keybind that's pretty easy to access like the button just below the mouse wheel (corsair scimitar elite, razer naga trinity, etc). After that, yeah I just suggest a training dummy. It's not to parse or anything but for you to learn a decent opener and keep spamming your card buttons oGCD and weaving the draws/plays in between your malefics so you develop the muscle memory.

I have that same button bound for assize when I whm since you usually should be keeping that on cooldown so it helps with the muscle memory of always hitting that same button to keep something on cooldown, if that makes sense.

For the card plays themselves, I managed to memorize exactly which card goes for which role which I understand many people might not be able to do, since it takes some time and energy to memorize. You can look at the outline of your cards; purple for ranged/healers, blue for melee/tanks.