Green’s Dictionary of Slang

to run away, to leave; to walk.

1836		Chester Chron 30 Dec. 4/1: A Yankee malcontent [...] Paddle off at once.
1838		R. Nicholson Cockney Adventures 6 Jan. 74: Jem Spriggins informed Tom Gubbs [...] that he must cut his stick. To which piece of informtion Tom Gubbs rejoined, in steam metaphor, that he must be paddling.
1840		Flash Mirror 6: Queering of a Duff Shop. — Going into an eating house, calling for a go of soup, prigging the knives and forks, pocketing the saltcellars [and] seizing a roll of duff, and paddling off scot-free.
1848		North-Carolinan (Fayetteville, NC) 18 Nov. 1/6: To leave [...] paddle.
1850		Man of Pleasure’s Illus. Pocket-book n.p.: ’Pipe this donna and swell paddling here’.
1853		Dickens ‘Slang’ in Household Words 24 Sept. 75/2: To go or run away is [...] to mizzle, to paddle, to cut, to cut your stick, to evaporate.
1862		J.A. Hardwick ‘The Browns Ruralising’ in Prince of Wales’ Own Song Book 41: She paddled off, of care no reck.
1873		Sl. Dict.