USA here. To me "call around" means make multiple phone calls. "I'd love to go, but let me call around and see if I can get a ride/date/babysitter."
I'm not sure "swag" and "swagger" are related. According to the dictionary, "swagger" means "walk or behave in a very confident and typically arrogant or aggressive way." "Swag" has multiple meanings ranging from "ornamental festoon" to "goods taken by a thief," but in modern American usage it's basically "stuff," generally free stuff with some promotional strings attached.
When I got replicas of Dr. House's cane and that oversized tennis ball he keeps in his office, that was swag, because they were free (Fox sent them out to us entertainment media folk). The bags of goodies they hand out to the stars at awards shows are swag -- again, because they're free. If we bought it ourselves, it wouldn't be swag.