Will,
From your description it sounds like there is no spring tension in either the trigger or the curved lever which sits behind the trigger. (ie. the trigger is flopping backwards and forwards).
If this is the case, it may be that the entire needle actuating assembly is not screwed in far enough. (This was a problem I have seen "out of the box" with one of these Badger 150 clones)
With the handle off, you from the extreme end (going forward towards the tip of the airbrush), you should see the following parts in the following order,
1: the rear-end of the needle
2: the needle lock nut (knurled brass)
3: the collar into which the needle lock nut is screwed (smooth brass)
4: the spring retaining screw (knurled brass)
5: The inner "barrel" which holds all these parts (in the diagram linked above, this is labelled as part number 50-030 "Tube shank", the outer end of which is knurled brass)
If part number "5" listed above isn't screwed in far enough, then the needle assembly simply will not move, the trigger and the curved "spring" lever will flop backwards and forwards. Screw it in (clockwise) and see if it engages the trigger mechanism. (You will have to eyeball this as there is no "stop" to let you know you've screwed it in far enough too far and it will lock the mechanism up solid, not far enough and it won't engage)
(BTW: I hope you didn't pay too much for it, from what I can see it looks identical to my $30 Super Cheap Autos airbrush) with a silver handle instead of black.