Rona, Being CUT does mean being fully drunk and HALF CUT does mean half so.
Cassell’s Dictionary of Slang
CUT adjective [18th century and still in use] drunk [abbreviation for CUT IN THE LEG, a facetious euphemism for being staggering drunk]
CUT IN THE LEG phrase [late 17th century to late 18th century] 1.very drunk 2. pregnant [[!!! hmm]] (cf. ‘cut in the back’)
CUT IN THE BACK phrase [mid-17th to mid-19th century] 1. very drunk 2. pregnant (cf. ‘cut n the leg), [‘cut’ adj.]
HALF CUT adjective 1. [late 19th century and still in use] (Australian) foolish, silly 2. [20th century] more than mildly drunk but not yet incapable (cf. ‘HALF-COCKED adj.; HALF-HIGH). [Standard English ‘half’ + CUT adj.]
HALF-COCKED adjective [20th century] mildly drunk (cf. ‘cocked’; ‘half-cut’ adj.; ‘half-gone’; ‘half-shot’ adj.; half-slewed) [[note plus the usual but unrelated meaning as in ‘go off half-cocked’]]
COCKED adjective[20th century] (U.S.) drunk (cf. ‘half-cocked’ adj.). [Standard English ‘cock,’ to pull back the hammer of a gun]
Ken Greenwald (Fort Collins, Colorado – U.S.A.
Reply from Ken Greenwald (Fort Collins, CO - U.S.A.)