Green’s Dictionary of Slang

limber adj.

(US) drunk, also used adv.; the image is of physical sloppiness, lack of co-ordination.

[US]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.
[US]P. Foster Pops Foster 82: Buddy and I used to get limber drunk on the bandstand. We couldn’t even stand up.

In compounds

limber leg (n.)

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

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