New Airbrush, leaks around Master Airbrush Adapter and Airbrush

Sounds like you need to get a correct adapter to match your hose to the airbrush or compressor. Not all adapters are the same. Check out www.tcpglobal.com. They have adapters. I tried looking for an adapter elsewhere but had no luck. You're better off going to TCP Global website instead.

I frequently do find leaks at fittings.  I now use teflon plumbing tape on all my connections.  Works great!

Another possibiltiy is the threads may be worn out too.

I always use some teflon tape on my AB/compressor/hose fittings too, cheap, easy to find, great aid.

Some adaptors I have ordered worked terribly, in some cases the thread pitch was way off, others worked perfectly. Buyer beware.

The threads are tiny, not robust at all, so I tighten minimally just until snug, relying instead on the teflon tape to provide additional seal efficiency.

Patrick