Like you, I used to be quite lackadaisical about cleaning my spray gun until a hangar neighbor showed me how to get better results. First of all, clean the gun after every use. Lacquer thinner works well but in your case with all of the hardened paint, you need to get some "Naked Gun" spray gun cleaner. Take the gun totally apart and soak it in the solvent. It's quite caustic so use thick gloves, not the latex or nitrile ones. It eats right through them. My friend also loads up on chopsticks when he eats Chinese. Sharpens them in a pencil sharpener and uses them as a pick that won't damage precise components of the spray gun. My DeVilbiss gun came with a cleaning kit with various picks and brushes to use on all of the different orifices. I have found it doesn't pay to clean up all of the rest of the mess and ignore the gun.
Naked Gun is available at just about any store that sells spray gun equipment.
Home Depot and Lowes don't.