I've never heard of a drinks only AI...food only, yes, but not drinks.
Smaller more affordable AI options in Negril include Legends as mentioned, Samsara, Fun Holiday, Charela, Merrils...mid range would be Grand Pineapple.

Another option would be to get a room with a fridge and make your own drinks.
It's always a blast to wander around to the different bars on the beach and cliffs...drinks overall are very reasonably priced in Negril.