Well.. I'm going to guess here. I'm having a little difficulty figuring out the precise events here. So with this 1:6 figure - Dragon/GI JOE size, painting the flesh to me is still a small area. A siphon feed is just fine!
Issue 1: Airbrush function - is your paint feeding better? I use siphon feeds all the time, they are not a limitation for this type of work. Please verify you're using new siphon cups or that these are perfectly clean. If there is resistance to the upflow of paint and the brush is having to draw paint through a high resistance system, the result will be inconsistent, sputtering & start/stopping. It sounds like things are feeding a little better with the new needle/nozzle set. For this size work 0.3-0.35mm nozzle and needle sets would be good. 0.2mm etc are for fine lines. If flow is improved, this is good. Milk consistency or little thinner (eyeball...) should work for ~0.3mm nozzles.
I noticed you're using Acryl. I've used that paint successfully for all sorts of painting in the past. Tips: Stir or shake the crap out of that stuff. Acrylics, esp Testors settle quickly, like enamel "sludge." Sludge no go through semi-clogged cups or 0.3mm nozzles. I have a paint shaker. Some people chuck bent wire into a dremel on LOW SPEED! But the paint needs serious mixing for upclose and personal airbrushing like this.
Problem #2: Acryl is NOT like an enamel. Thinners for acylics lower VISCOSITY. They also tend to hasten DRYING. Something I have noticed. Tamiya is great at that game too.
It sounds like you are attempting to apply wet-on-wet coats of paint. Acrylics, unlike enamels tend to dry in a shell/film and don't really like wet-on-wet. Disturbing the undercoat with high pressure new paint will peel the layers and make a very inconsistent finish. If you are fast, paint the one shade you want. Let cure for ... 5-10min...total wild guess. Hot lights can help. Then low - 10PSI with well thinned paint could be applied for highlights etc. Between coats, I would totally flush the brush with soapy water, then water and leave water in the brush. Add/change your color in the cup then start the next coat. I'd only try this like 2-3 times in a session. I don't know how many coats you want to apply.
I do not have exact experience with Acryl but I would attempt to use a retarding agent that slows drying in the base color so it wont form clots on the tip of the brush and work more like enamels. Liquitex, GOLDEN, Amsterdam? are available. Liquitex should be ok, I use GOLDEN. Re-reading the post- the wet mess sounds like too much paint. Acylics are all about layers. I prime large scale figures in BLACK! It may take 10+ thin airbrushings of acrylic for the base color alone. I use a retarder in the base color, take the bottle in and out of the brush, it doesn't clog up that quick (if sealed bottle). I use vallejo these days, acryl in the past and they take about 10min for a good cure, then overcoat. DON'T rush the drying, it can turn into the fluid mess. These may help with your wet-on-wet applications.
Summary, I don't think it is feasible to base coat in thin layers, shade with a couple colors and highlight with colors in one session. I would get a fantastic base coat in thin layers. Leave for day, maybe seal with a good semigloss varnish to protect from future applications. Then do shadowing/highlights. Acrylics (especially aibrushed) don't blend wet-on-wet like enamels, oils or water-miscible oils (brush techniques). The "blended" effect on acrylic figures is more about what you actually are doing which is multiple different shades applied in succession. 1:6 is going to require lots - but that is from the expert miniaturist side of the world - NOT ME! I'd like to see a nice basecoat, simple and effective shadows and highlights executed very well. No, it won't look like some of the insane box-art from Andrea miniatures... but it won't look like a messy attempt at what the experts do.
This is learning new paint, new chemistry on a difficult subject - a figure!
best of luck.
