Green’s Dictionary of Slang

dragon n.4

(US) a (large) penis.

[Lauging Mercury 8-16 Sept. 184: A Levites Wife was this last week taken encountering with a Dragon at the George tavern [...] [...] being stung very happily in a place that nothing endangered her life.
[[UK]J. Phillips Maronides (1678) VI 111: St George he takes a furious course, / The Dragon spits, away flies horse, / Leaving St George upon the grass, / The sport of many a pretty Lass].
[US] P. Munro Sl. U.
[SA]K. Cage Gayle 67/1: dragon n. gigantic penis.

In phrases

drain the dragon (v.)

see under drain v.