I wouldn't soak the air valve assembly in any sort of solvent.
You should be able to unscrew the air valve from the airbrush body as a complete unit.
When you do this, you should be able to see the upper end of the plunger with an o-ring around it - I suspect that this o-ring may have gummed up. Refer to part #19 in the diagram below.
Carefully slide the o-ring off the plunger stalk (don't lose it - it's only 3mm across and less than a millimetre thick) and wipe it with a tissue - if it is sticky or it leaves a lot of black residue on the tissue, (or if there's sticky black residue on the plunger stalk) then it may be perishing - if so, there's not a lot you can do about it except replace it - and I don't know where you can get a replacement o-ring of that size.
If it's not perishing then you're ahead of the game. With the o-ring "off" the plunger, press the plunger down and see if it returns smoothly. If so, re-install the o-ring and hit it with a little silicone based lubricant. It's important to use a silicone based lubricant and not a petroleum based one (which will, in the worst-case melt the o-ring, or make the o-ring swell, causing the plunger to stick again)
This is not the exact same model you have, but it should be close enough for this purpose:
