Plain water is actually pretty high viscosity compared to well-thinned alcohol or spirit-based paint. Try turpentine instead. The A270 has more problems than any other airbrush I've seen. Try both nozzle positions, and this is a stupid question but be sure you have the nozzle screwed open far enough. Open it until paint comes out. Being an external mix airbrush it may be hard to get started, try twisting the cup so it won't spill and turning the airbrush on its side so gravity can help get the paint moving.
I would also recommend you switch to Tamiya acrylic until you're ready to buy an "expensive" airbrush, because most enamels do not spray well through plastic airbrushes. acrylics are in essence liquid plastic themselves, so they work better. Enamels can do very well in a good airbrush, and their colorfastness in definitely superior. But they just don't seem to want to spray from a plastic airbrush.
Also the PSI shouldn't be an issue, with my Badger 100 I spray from 3 or 4 PSI all the way up to 80 or 90 sometimes.