Green’s Dictionary of Slang

rince v.

to drink.

[UK]Awdeley Fraternitye of Vacabondes in Viles & Furnivall (1907) 13: Rince Pytcher is he that will drinke out his thrift at the ale or wine, and be oft times dronke.
[UK]Jack Randall’s Diary 34: Bid him not turn ’em up for the rincing [...] tell him, my oldone, without any mincing, You mopp’d them out (*drank up the remainder) ere you brought them to him.