What kind of paint are you using? Laquer, enamel, or acrylic? Acrylic dries so quickly that it can partially dry between the airbrush and surface.
How far from the surface are you spraying? The farther away the more "Feathering" you'll get on the edges. If you are more than a couple of inches from the surface try thinner paint, lower pressure, and closer to the surface. Those three parameters (pressure, viscosity, and distance) are related so you may have to experiment to find what works for you.
If you can lean the airbrush so that it is pointed into an area slightly it will reduce the overspray on one side.
Do you have a moisture trap? If the humidity in your area is high you may be getting some moisture through the brush.
Airbrushes spray in a cone pattern and do not give a perfectly hard delineation between colors. There is always going to be some feathering on both sides of the line. If you want a hard line you will have to mask it.