Green’s Dictionary of Slang

1. (also dropsey) one who ‘drops off’ to sleep.

1836–7		Dickens Pickwick Papers (1999) 365: Come, vake up, young dropsy.
1866		Night Side of N.Y. 118: And on the other hand ‘young dropsey’s’ legs and arms were like links of dryed ‘bolonas’ in the garments which misfortune’s raffle had drawn for him.

2. (W.I.) an illness whereby the sufferer is prone to sudden sleepiness.

1995

Francis-Jackson Official Dancehall Dict. 16: Dropsy illness relating to anyone who suffers or is prone to falling asleep almost anywhere at anytime: u. de bwoy ’ave dropsy.