swag (v.)youdaoicibaDictYouDict
"to move heavily or unsteadily," 1520s, probably from a Scandinavian source akin to Old Norse sveggja "to swing, sway," from the same source as Old English swingan "to swing" (see swing (v.)). Related: Swagged; swagging.
wag (v.)youdaoicibaDictYouDict
early 13c. (intransitive), "waver, vacillate, lack steadfastness," probably from a Scandinavian source (compare Old Norse vagga "a cradle," Danish vugge "rock a cradle," Old Swedish wagga "fluctuate, rock" a cradle), and in part from Old English wagian "move backwards and forwards;" all from Proto-Germanic *wag- (cognates: Old High German weggen, Gothic wagjan "to wag"), probably from PIE root *wegh- "to move about" (see weigh).

Transitive meaning "move (something) back and forth or up and down" is from c. 1300; of dogs and their tails from mid-15c.: "and whanne they [hounds] see the hure maystre they wol make him cheere and wagge hur tayles upon him." [Edward, Duke of York, "The Master of Game," 1456]. Related: Wagged; wagging. Wag-at-the-wall (1825) was an old name for a hanging clock with pendulum and weights exposed.