Green’s Dictionary of Slang

1. worrying.

1884

Sporting Times 1 Nov. 2/5: At the closing of the year / I now find things rather queer / And that times may soon be what is known as ‘breezy’.

2. frightened, fearful.

1921		(con. WWI) E. Lynch Somme Mud 67: They’re getting breezy that Fritz is putting gas over.
1925		(con. WWI) Fraser & Gibbons Soldier and Sailor Words 36: Breezy: Nervous. Easily frightened.
1940		F. Sargeson ‘Three Men’ in A Man And His Wife (1944) 69: I was so hot I had to fan myself with a hanky, and Hilda saw me and said I must be getting breezy.
1950		A. Gurney Bluey & Curley 1 Jan. [synd. cartoon] Fair dinkum, I’m breezy handling these cases [i.e. of high explosives].