Green’s Dictionary of Slang

(drugs) to take a drug, e.g. cocaine or cannabis.

1934		F. Tuohy Inside Dope 102: One can get the stuff so easily from Marseilles, and draw and puff immune from interference.
1992		V. Headley Yardie 12: So wha’, you a draw too?
1999		Indep. Rev. 19 Aug. 5: The basic social [...] event of smoking-chuffing-drawing a few joints-spliffs-cones is largely taken for granted.