Ok, here's what the problem is, based on what I am reading.
You get air through the airbrush, it comes out under pressure, so you know that this is all working ok.
You put your finger over the end, and the back pressure will cause bubbles in the reservoir, so you know that in reverse the air goes to the paint.
You take your finger off and depress the trigger, and nothing comes out, so you know that the air is going to the needle tip, but it isn't sucking paint through.
This tells you where the blockage is.
I would hazard a guess that you have a lump of dried paint stuck in the passage between the paint reservoir and the inlet for the air.
When you use back pressure it acts as a one way valve, it pushes the blocked paint out of the way, and lets air back into the reservoir, take your finger off and the valve closes, blocking off the paint supply.
If you remove the needle, fill the reservoir and nothing drains out it pretty much confirms where the blockage is.
There are two things you can do.
Remove the needle, fill the reservoir with airbrush cleaner and leave overnight.
Next morning use a cotton bud to scrub right down into the reservoir neck.
Only use back pressure to agitate the airbrush cleaner, otherwise you will be sucking the blockage further down the passageway.
Empty the reservoir by pouring it out, not by using the airbrush as normal.
I would also suggest using proprietary airbrush cleaner, rather than turpentine, or water. The real stuff really does make a difference. If you can get hold of some fine brass wire you can use this to help clear out the blockage.
Once you have fluid coming through the airbrush just keep running it as normal, using back pressure, and emptying the reservoir by pouring the contents out. Eventually you will get rid of the blockage and any residue, but if it is solidly dried paint, it will take some patience to get it out.
I thought my airbrush was trashed, or my compressor was faulty, or the regulator was faulty when mine did this but sure enough it was paint build up. I now run cleaner through the airbrush and leave it with some left in the reservoir overnight whenever I use it. I only use diluted acrylics and I get this problem, if you have hardened enamel down there you may never get it out, but its worth a try.
The other alternative is to buy a Chinese airbrush, I have one, as cheap as chips. It has lasted me years, but at least I know that I can bin it and buy a new one without breaking the bank.
I have the bottom one in this pic. If you get one with adjustable airflow on the brush, you can control pressure from the brush as well as the regulator, which is great if you don't have a regulator, or you want fine adjustment while you are spraying.
https://www.everythingairbrush.com/airb ... h-kit.html