Congrats, JazzyJ, and welcome aboard from a fellow Alabamian.
I agree that you aren't required to show your concealed-carry permit (often referred to in my County as simply a "pistol permit") if involved in a traffic stop here; however, in the handful of instances I've been pulled-over, I've always handed over both my driver's license and permit and have advised the officer that a firearm is present - I've always been thanked by the officer and, once, even got into a "what-are-you-carrying-oh-that's-cool" discussion ha-ha!
I'm glad Alabama is such a predominantly "gun-friendly" state. Also, as you may already know, we share reciprocity with a good number of other states with regard to concealed-carry. Check out the State Attorney General's website for more detail:
http://www.ago.state.al.us/Page-Gun-Reprocity-Law%20
As for competitive shooting, opportunities abound. If you're near Birmingham, have a look at the Brock's Gap Shooting Facility in the Hoover area. Among other things, they have a thriving IDPA element:
http://brocksgapidpa.com/
Feel free to send me a PM anytime and again, welcome aboard!