You didn't say what brand of paint you are using, but from the remark about running water through your airbrush, I assume you use acrylics. Don't use Tamiya or Gunze Sanyo, or any alcohol reduced acrylic. Use only water reduceable types (Polly Scale is a good one).
Secondly, use an acrylic retarder, probably about twice the normal amount (two drops per thinned airbrush cup) and reduce the paint more than normal. I can't give you a specific recommendation for thinning ratio or consistency—you'll have to experiment. You will probably have to apply numerous very thin coats.
Finally, airbrush at the lowest possible pressure, again, you will have to experiment, but aim for 10 psi.
As for the airbrush, keep a squirt bottle of Windex handy. As soon as you are done, dump any remaining paint, wipe the cup, and blast Windex through the airbrush. Rinse with water.
Tip dry will probably be a problem. A cotton swab in a small container filled with Windex, then applied to the tip each time you stop (and probably occasionally while spraying) should clean the problem up.
Good luck!