If you go the standard permit route, you apply for your permit, wait a week, and go back to pick it up. If you go the CCW route, you will need to wait for a class to open up, take the class, submit all your paperwork, and then wait a minimum of 90 days for the permit to come in. My advice is to get get both; the regular permit for $5 so you can get your gun and practice with it while you wait for your CCW class and permit to arrive.
In regards to moving, CCW permits are not transferable as I know; you can't turn in your NC permit for a TX one, as an example. You are probably going to have to start the whole process over again. If you know where you're going, you may want to find out if you can get some sort of non-resident permit and if you can convert it to a resident permit once you establish residency there.