Welcome to OCOD from a neighbor to the north. I spent a couple of summers and a semester in Tucson while completing my school work some 20 years back. I'm now in the Salt Lake City area.
There are a couple of recent threads on this very topic; though neither is specific to Arizona.
Apparently, if you read through the fine print of your Costco membership agreement, all privately carried guns are banned.
In Utah, such private gun policies at businesses have no force of law so I've never bothered to look very carefully for where in the agreement this language shows up. I just untuck my shirt or put on my jacket (depending on the weather) and CC or casually CC while in Costco. My gun has never been noticed, at least never mentioned.
Others choose to boycott Costco.
At the very least, you should double check your State laws and make sure that carrying in a business in violation of a poorly notified "no gun" policy doesn't violate any laws before carrying into such a location.
All the best.