I feels your pain.
"Despite this, I usually will have to take my airbrush completely apart and put it back together to get it working." This seems like you are doing a pretty good job of cleaning, but there is some obscure choke point in the paint path that is eluding your cleaning regimen.
In my case there was a thin coat of paint inside of the nozzle(AB003 in the diagram below), that I swore I was cleaning out. Once I new to watch out for it, I started getting more consistent success from my airbrush.
When you first break out your airbrush, run some cleaner or even water through it. You should be able to see a consistent flow. Then work your way up with properly thinned paints. Remember usually multiple thin coats is better than one thick one anyway.
Sounds like par for the course for the type of compressor you are using. I gave up on mine for the noise and the fact that I would see fluctuation in the pressure being supplied as the compressor turned on and off.
Now I just use a $100 craftsman compressor that I use with my nail gun.