Green’s Dictionary of Slang

(US) a sour-faced person, a grumbler, a killjoy; also attrib.

1935		M. Fulcher ‘Believe Me’ in Afro-American (Baltimore, MD) 13 Apr. 12/1: Drizzlepuss Winchester doesn’t even know that [a song] was written by Reggie Forsythe.
1938		L.A. Times 17 July This Week Mag.4: [headling] ‘He’s an opinionated, fat old drizzlepess.’ That was the locker room’s verdict.
1943		‘Whitman College Sl.’ in AS XVIII:2 Apr. 154/1: drizzle puss, drone, drool, drip. These vulgar monosyllables are applied to anyone not up to par socially.
1949		Age (Melbourne) 2 Apr. 7/2: His fiance is Ann Freeman, a colorless damsel with a drizzle-puss expression.
1950		Hal Ellson Tomboy (1952) 93: Who caught them [i.e. bullets] for him? Not drizzle-puss Mary.
1954		Panama City News-Herald (FL) 3 Aug. 3/5: It’s always the homeliest men who have the loveliest wavies, look at old drizzle-puss Gavin there, and Lilith — .
1955		Ottowa Jrnl (Ontario) 18 June Sat. Section 37/4: Nobody likes a drizzlepuss or a deadpan as people, and no one likes them much better in pictures.
1966		Nevada State Jrnl (Reno, NV) 16 June 4/3: ‘For two years I have been trying to find a human way to get rid of my dizzle-puss secretary’.