I sweat when I walk about. Is this an indication that I'm out of shape?

No, not necessarily. Everyone has their own sweat rate. Some of us sweat more than others. Obviously, your activity level and environmental factors, eg the temperature, can also affect how much you sweat, but your sweat rate remains essentially the same regardless of your fitness. However, it may mean that you are out of shape if you are sweating because that low level of activity represents a strenuous effort.