Yes, Azamara does provide a free shuttle in most ports where the ship docks away from the town. There are some exceptions in places where the local taxi union has an agreement with the port authority not to allow this. However because Azamara ships are small they are often able to dock close to the town.
We always buy a full Wifi package as the Wifi speed is not brilliant and you end up wasting precious minutes logging on and off. The Wifi is in the process of being upgraded across the ships, so this may change if the speed increases.