Green’s Dictionary of Slang

(US tramp) a chase, a hue and cry; a ‘tail’.

1927		F.M. Thrasher Gang 94: ‘Sometimes we’d raid ice-cream wagons. If the driver saw us, he would jump off and give us a shag’.
1931		Irwin Amer. Tramp and Und. Sl. 167: Shag.–A chase or organized pursuit by the police or by an irate citizenry.
1948		C.S. Montanye ‘Crepe for Suzette’ Thrilling Detective Oct. 🌐 When Mullin thought I was chiseling on the police he had a habit of sticking a shag on me [...] But nobody tailed me to Broadway.
1962		Ragen & Finston World’s Toughest Prison 817: shag – A chase or organized pursuit by the police or an irate citizenry.