First things first, learn to wear your shirts untucked. It's possible to carry around a tucked in shirt, but it limits you so much it's just easier to change your style to open up more options.
I am almost exactly your size and my everyday carry is a Glock 26. In a good holster it conceals fine under a loose t-shirt or polo (I wear a tighter undershirt so the gun is not against my skin)
In the past I have also carried .38 and .357 snubbies, and a Glock 23 with the same results. In your price range you should have no problem getting a gun, a GOOD holster, a sturdy belt (this is a must), and an extra magazine.
If you are stuck on a single stack for more concealability I have shot and handled a Ruger LC 9, they are thin and easy to conceal, but I didn't like the trigger pull too much. Don't rule out a small revolver, they have been serving people well for a long time (I really miss mine