Green’s Dictionary of Slang

limber adj.

(US) drunk.

1860

Spirit of Democracy (Woodsfield, OH) 25 July 4/1: Synonyms [for drunk] [...] slewed, fuzzled, [...] swamped, raddled, nappy, [...] having a brick in one’s hat, limber, tired [...] toddled, slung-shot.

In compounds

limber leg (n.)

(US black) impotence, caused by drinking jake n.³ (2)

1934

Willie Lofton ‘Jake leg Blues’ lyrics] I said I drunk so much jake, ooh Lord till it done give him [sic] the limber leg.