as far as i can think of there's no concise word/term for it so i'd just say "a drink that doesn't quench thirst" or something like that
Hm... not that I'm aware of, but I understand your situation, both with unquenched thirst and awkward situations like that lol. I usually just describe it as "It didn't quench my thirst." Nothing too special.
Adding on to "still thirsty," I'd consider it a safe backup in the event whoever you're speaking to doesn't know what "quench" means.