Green’s Dictionary of Slang

damp v.¹

to have a drink.

1820		‘Peter Corcoran’ ‘Gloss.’ in Fancy 109: damp – To wet with heavy brown, or stark-naked.
1837		‘Darby the Swift’ in Bentley’s Misc. July 71: We damp’d the grief a trifle at the shebeen with a drop of the rale stone turf.
1846		Swell’s Night Guide 117/2: Damp your mugs, wet your mouth, drink.
1908		Denton (MD) Journal 7 Mar. 3/8: ‘You come and damp it.’.

In phrases

damp the dust (v.) (also dampen the dust, damp the sawdust)

(US) to take a drink.

1851		G. Thompson Jack Harold 57: Laughter and applause, during which Bill ‘dampened his dust’ and renewed his quid.
1890–1904		Farmer & Henley Sl. and Its Analogues II 250/2: Damp the Sawdust [...] (licensed victuallers’) To ‘crack a bottle’ with friends ‘for luck’ on starting a new ‘house’.