Green’s Dictionary of Slang

busting adj.²

desperate.

1874		‘A.P.’ [Arthur Pember] Mysteries and Miseries 360: [S]pringing up, he called out: ‘Boss, I’m bustin’ to make a speech. Somebody run for a wooden chair for me to stand on’.
1963–74		B. Humphries Barry McKenzie [comic strip] in Complete Barry McKenzie (1988) 15: If I wasn’t bustin’ to splash the boots I’d do this pom over.
1968		G. Slatter Pagan Game (1969) 175: The men busting for a leak.
1979		Lette & Carey Puberty Blues 2: We were also busting to go to the dunny, but that was too rude for girls.
1991		R.G. Barrett Between the Devlin 76: Les suddenly found himself absolutely busting for a leak.
2016		N. Cummins Adventures of the Honey Badger [ebook] I’d never been so busting for a holiday.