You are mistaken regarding the content of the law. The law, as passed, allows (and even requires) police to detain anyone they reasonably suspect is in the country without authorization, e.g. a person the police encounters during a traffic stop. As far as I know, the text of the law does not define what constitutes a basis for a reasonable suspicion that someone is an and does not explicitly address how exactly people are supposed to prove their legal status. A driver's license from another state does not really conclusively prove legal status or citizenship since different states have different standards for issuing licenses and for determining their periods of validity. As a practical matter, hopefully the police officers in Arizona will exercise a bit of common sense and will not detain someone with a valid out-of-state license. But this is really up to their discretion.
Another provision of the Arizona law requires all legal aliens, both LPRs and those on nonimmigrant visas, to carry their immigration documents with them at all times. (Although I think that provision is currently stayed by the courts).
Like I said, I think the law in blatantly unconstitutional on several grounds and it is a great shame for Arizona that they enacted such an absurd piece of legislation.
You don't show true patriotism for your state by blindly defending anything they do. Instead you should by out there, fighting to convince the Arizona legislators to rescind that law and to respect the Constitution.
I have lived in Illinois for over 10 years, but I am far from being beholden to defending everything that Illinois does. On the contrary, Illinois also has plenty of things to be ashamed of, e.g. an especially corrupt political culture.