Or do the rails need a load to properly regulate?
Yes, most computer switchmode supplies are designed to work with a minimum load. Without that, it's normal to see output voltages somewhat out of range as you're seeing.
Things are pretty close, suggesting that nothing is critically unhappy, so you're not going to damage the A1000 by connecting it.
Under load, the various outputs should be within about 5% of the nominal voltage. i.e. 4.75-5.25V for the 5V line.
