Green’s Dictionary of Slang

(US) to move carefully, quietly.

1887		Scribner’s Mag. 2.512: It is said that the audacious among ‘bullwhackers’ dance from this elevation [i.e. in high-heeled boots], but only he can believe it who has seen them egging around in a doleful bolero [DARE].
1950		WELS n.p.: (To get out of an uncomfortable situation: ‘They thought they had him cornered, but he managed to ________’) [...] Egg his way out [DARE].