What the achievement asks
Acquire twenty-two Kul Tiran drinks. Five points, account-wide.
Twenty-one of them are simply bought from nine vendors, spread across Tiragarde Sound, Drustvar, Boralus and Stormsong Valley. The twenty-second is not sold at all.
You only have to acquire them, not drink them. The credit lands when the drink enters your bags.
The round
Ten stops, in an order that crosses each zone once:
Two of them carry half the list on their own: Sarella Griffin at Norwington Estate sells six, and Linda Deepwater at The Drust Bar sells four. Start with those two.
The twenty-two
The one that is not for sale
Long Forgotten Rum comes from a cave in Deadwash, in Stormsong Valley, not from any vendor. Looking for a barkeep who stocks it is time wasted.