When using my airbrush (Badger 150) I thin my paints to about the consistency of skim milk. I've been happy with the results thus far. My first project was this land raider and the airbrush was used for the Mordian Blue basecoat and the brown and black washes that were used for the initial weathering/shading.
My third project (no pics of my second but it was just shading on some terrain) was this wave serpent, using the airbrush for the fade from black to blood red, then the highlighting and the rest of the fade was done with a brush.
The things I've learned so far: Never underestimate the importance of thinned paint! Not only does it go on smoother, but with a double action airbrush you will find you have better control of the paint flow. I've encountered the same thing you have with the dry tips, but it is noticably reduced running very thin paint. I have ordered some liquitex flow improver and drying retarder to try out in the future, but I think the dry tips are just going to have to be dealt with by a little cleanup now and then. I cleaned mine out whenever I needed to switch colors and that worked fine. There's a fine line between thin enough and too thin though, as you want to maintain an even coat and not turn your paint into a wash.
If you have hard water in your area, I'd recommend thinning your paints with distilled water. This recommendation applies for paints applied with a regular brush too but makes a bigger difference with an airbrush.
The best thing to do is to try it out, as paper is much more absorbent than a model and isn't going to give nearly the same results. I was surprised at just how forgiving it was the first time I tried it. I haven't tried airbrushing over white primer yet, but I think black will be easier to learn with as your colors won't build quite as quickly. Both of the models in my photos were GW black primed.
I hope this helps and best of luck!