Bit late on this one, sorry, (been fixing an annoying v slow water leak in cellar!) but maybe my own quite extensive experience of airbrushing (mainly for flying model aircraft - NOT woodwork) plus lately a bit of a mess about with a wooden toy tractor may help.
1st entirely agree with the others that for the sort of coverage you need you don't really need a double action AB, but now you've got it they really are the "best"(i.e. most potential capabilities), BUT you have to learn how to use it properly, and just as Brandlin has said, that takes PRACTICE. Then some more (practice)!
In fact I guess you could get by with only a single action/external mix brush (I use a Badger 350) for what you need but now you've got the double action brush (mine is a Badger 150) just stick with it because yours will definitely produce a finer line (and give you more "gradual graduations") than, say, my 350, even if fitted with the finest nozzle/needle. But (sorry again) it really does take practice - and to make matters worse, you'll find you need slightly different techniques and mixtures according to what type of paint you're using.
Did your AB come with a "Manual" complete with practice exercises? If not PM me and I'll send you a copy of what I got (years ago now) with my Badger 150. They really did help me to "get it" with air brushing.
2nd, also entirely agree with Brandlin that 100% cleanliness is absolutely VITAL, and sorry, but if you've used the AB once then it MUST be cleaned. You can get away with "flushing" the AB between colours during the same session (pure air + thinners and change the cup/bottle), but as soon as the session is finished then it's a complete strip, clean, and reassembly (careful with the needle)!
3rd Custard is dead right, temperature plays a big role when spraying (as does humidity, especially if spraying any sort of quick drying colour like cellulose). Personally I would not consider spraying any sort of paint or colour at all if the ambient temp in the shop is less than 10 deg C.
4th I think 30 psi is much too high and may be one possible reason why you're getting spatter (see item 5 below). Does your compressor allow you to adjust pressure? If so I SUGGEST about 15 to 20 psi should be enough. If not adjustable (like my Badger compressor)then get an extra long line to go between the compressor and the AB. Also, make sure to get the thick-walled type of line as both the thickness and the over-length line significantly reduce pressure pulses AND reduces output pressure at the AB itself, especially if there's no reservoir and pressure regulator on your compressor - probably not?
5th has your compressor got an oil/moisture trap between the compressor and the AB line? If not I highly recommend you fit one - this MAY be another reason why you're getting spatters (even if "only" using water-based paints/colours).
6th also highly agree with Droogs about getting a viscosity cup. If you're not familiar, this is basically a bit like a funnel with graduation/s (line/s) at the top and a certain size outlet at the bottom. To use, you mix up whatever paint and thinner then pour into the cup up to whatever line you choose (finger over the outlet) then open the outlet and time EXACTLY how long the mixture takes to pass through the cup. I doesn't matter which cup you use (I use a Ford, I think off hand it's a No. 7), but the point is you make notes of all the variables (different paint/colour type and brand, % mixture, quantity mixed, temperature) each time you use the AB so that you soon build up a "data base" of exactly what works for you under all conditions.
7th I'm not sure what medium you're using, but when I did the toy tractor I used artists acrylics for the first time (child safety and all that) and that stuff comes in a tube and is about as "fluid" as toothpaste)! I had to mix it down to 70+%, aiming, just as the others have said, for a consistency like milk (i.e. just a bit thicker than water - but NOT much thicker). To that I added a TINY drop of washing up fluid (syringe). You can use car windscreen washer additive instead. The idea of either of these is to reduce the surface tension of the mix when it's sitting in the AB bowl/cup
From what Custard has been saying above, it looks like your colour medium is a LOT thinner than the above acrylics I used to start with (!!), but using the "like milk" advice as a basic visual guide, that, added to the viscosity cup, will get you to where you need to be.
In short, the % mixture, along with an even slightly dirty AB (the orifices in double action brushes are REALLY tiny), line pressure, and water contamination are the other most likely reason/s why you were getting spatters on the job.
Sorry this is so long, but as the other posters have already indicated, it's a subject with a number of variables just hanging around to trip up the unwary.
HTH anyway.
With a double action AB you really do have the "best" tool possible (no matter what brand I think - it's what all the real airbrush artists use) and once you master it "the sky's the limit" as they say¨.
Good luck
AES
Oh yes, sorry. A quick P.S. Whatever medium you're using, a really thorough shaking/stirring before you start thinning for AB-ing is also absolutely vital. (In my model making days I used one of those battery-powered "joke-type" drinks mixer thingys. Really a joke those things, but VERY effective in making sure that no colour pigment was hiding at the bottom of the tin/jar).