Section 13A-11-74Has exceptions to needing a permit. Taking a handgun to the range is not allowed and people that do so are breaking the law if they do not have a pistol permit. At least, that's how it reads. I've yet to see a case of someone being convicted under 11-73 for going to and from a range.
In my experience, cops aren't going to care as long as it's unloaded an in a locker or trunk or something, but they probably could charge you if they were in a foul mood. Remember that federal travel law doesn't apply unless you are travelling interstate.
Inside a vehicle is not considered 'concealed'. The law simply states that carrying a handgun inside a vehicle or carrying a concealed handgun requires a permit.
Easiest thing to do is just a get a permit. There isn't a residency time requirement in Alabama so as soon as you move here you can go apply at the Sheriff's. Turnaround time seems pretty fast here too. I had my app approved 2-3 days after i turned it in.
If you are concerned about not being able to carry, you should contact the Sheriff and see if you can file the paperwork before you move and be able to pick up the permits the day you become residents.