One other thing that I don't think I have seen mentioned. The spring tension on the needle chuck might be a bit loose. The Needle chucking guide screws in an holds the spring that gives resistance and pushes the needle forward in place. If this is a bit loose, which is common in shipping, and you use it straight out of the box, this is a likely culprit. It may not be pushing the needle in all the way. You can turn it clockwise to tighten and increase needle tension. That should make it press a bit firmer against the inside of the nozzle and prevent any paint leakage.
There are some great tips above. But more than anything I would stress being gentle. Firm but gentle. Nozzles are very, very easy to damage and a common pitfall for new users that are overaggressive in cleaning. If you disassemble, things should really only go back together finger tight. Don't use tools if you can possibly avoid it.
Airbrushes are tested at the factory, so it's possible, but unlikely that there is a trace of dried paint in the nozzle preventing a tight fit. Easy enough to deal with.... gently take out the needle from the rear, put the airbrush in some hot soapy water for 40 min to an hour rinse thoroughly and when re-inserting the needle, give it a couple turns, pull it out and look at the tip for any dried paint that's flaked off, wipe away and reinsert.
Once you have your brush assembled, test sprays with water to clear out any remaining junk and then let dry.
The number one cause of airbrush damage or weird performance for new users is over aggressive cleaning. I cannot stress this enough. It was an expensive lesson for me as a young artist. There wasn't an internet where I could look things up when I was young, so I learned the hard, and very costly way not to do that.
Basically, once you have a good spray with water, do your absolute level best to never, ever remove the air cap, or nozzle for cleaning. Avoid it as much as humanly possible. Use soap and hot water (liquid dishwasher soap is about the best thing ever for removing old paint), and soak. Never force anything. Avoid using nozzle cleaning pins, or anything that goes into the nozzle. The magic of hot water and soap is about all you should use unless there is absolutely no other choice.
With model paints, from Tamiya, Citadel, Army Painter, green stuff, etc... ALL of them need to be over reduced, especially the white or light colors when doing zenithal highlighting. The pigment particles are large and clog super easily. Beware their reducers however. Tamiya is actually a waterbourne paint, like most acrylics. Not an alcohol. Alcohol is added to their reducer to help it dry faster, but that can be a nightmare with an airbrush, leading to tip dry and clogs. I strongly recommend you look at reducing with something like Liquitex or Createx reducers, or roll your own with the classic recipe for acrylic reducer / flow improver made with glass cleaner, a dash of denatured alcohol, a lot of water and a teaspoon of glycerin. There are plenty of recipes online.
For your highlights, work in multiple thin, thin, thin layers. Dry fully between layers. Otherwise you will get paint buildup, it will take forever to cure, and you'll lose a ton of model detail.