I never take the fluid needle out. I take the air cap off. I take the nozzle off and put both in a small jar or thinner as long as its okay for that type of air cap. You should pull the trigger back when you take the nozzle off so it doesnt rub on the fluid needle.

Then I use an artist brush and just clean the head and needle, have a squirt bottle as well to squirt some thinner around. Then blow it out so its dry, May not be conventional but you take the needle out you wear the packing and more of a chance you may damage the fluid needle if not careful enough. Ive had zero issues with this doing this forever.