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Washes are never applied by airbrush, just by brush. Your problem with the shine and slow drying time stems from your use of natural turpenoid. As you have just discovered, you want to stay away from that. Here's some potentially really bad news: that wash may take a looooong time to dry, like days or even weeks (I've seen it said, no personal experience). The natural turpenoid is bad news. But at least you've crossed one alternative off your list.
Another tip for washes: If you choose not to clear coat your model before washing, try first washing the surface with pure, clean thinner. While still wet with that wash, apply the tinted wash. This will help the wash flow, much like a smooth clear coat. And regarding your use of acrylic basecoat with oil washes, that combination is fine. In fact, it is optimal because most any thinner you use for the wash will not attack the acrylic paint (exception being lacquer thinner, which has no business being used for a wash to begin with). Good luck, and move on to your next project, that natural turps wash is going to take its sweet time.