Green’s Dictionary of Slang

1. (Aus./US) tipsy, mildly drunk.

1818		M.L. Weems Drunkard’s Looking Glass (1929) 60: The patient goes by a variety of nicknames [...] such as boozy—groggy—blue—damp.
1895		Coburg Leader (Vic.) 8 June 1/5: The curly-haired ex-sec. of the Wicks got very damp in the third quarter’s play.
1910		Sun. Times (Perth) 30 Jan. 2nd sect. 4/7: A very damp visitor [...] deployed into the bar of the Shamrock from the office, aad held forth on the hotel clerk. ‘The (hic) cow wouldn’t change me cheque. Me (hic) ’oo’s got thousands in the bank’.
1919		A. Baer Two and Three 22 Feb. [synd. col.] ‘Doc, if you can make my dry cough a little damp, I’ll give all my friends your right address’.

2. foolish, stupid.

1908		A.N. Lyons Arthur’s 10: ’Ulking blokes. Meant all right, but slow an’ damp.
1944		R.L. Bellem ‘Focus on Death’ Hollywood Detective Jan. 🌐 You’re all damp. I’m sorry to hear you’re being blackmailed, but I’m not the shakedown artist.