Sounds like a double action to me, as well.
On a DA brush, you push down for air and pull back for paint, all with the same trigger. You can control the amount of air and paint depending on how hard you press down and how far you pull back.
Single action is one trigger that activates the air which draws in the paint (from either a cup or bottle), but you can't control the amount of paint. It's a direct correlation to the air pressure being used. More air pressure creates a higher vaccuum which draws in more paint.
I believe there is a hybrid brush out there that looks and acts like a single action. You push down on the trigger for air, but there is a control valve on the paint feed which allows you to control the amount of paint getting into the airstream.
As far as your other problem, it sounds like you are shooting way too high. Try dropping the pressure between 8 and 15 psi (different brush/compressor combinations act better at different psi). I personally like to shoot in the 8-12psi range depending on the area of the piece I'm shooting. Fine work requires lower psi.
Thinning is as much a skill as painting. Start at 50-50 and adjust from there. Some paints like 1:1 paint:thinner, some like 2:1, so it's a matter of trial and error. Some say look for a consistency of milk (or was it 2% milk, I forget).
Also, keep in mind that you won't paint the entire model with just your airbrush. Most models are done in various media - airbrush, paintbrush, pastels, oils even watercolor.
Well, hope that helps some!
-Fred