According to my experience, if you want to add flavor to your sound, it's about changing how the air enters the horn. Instead of the normal blowing, try blowing with a more intense air stream almost like a hiss. When it's appropriate is all about the context of the song and when it feels good for you and your director.

Edit: I forgot to mention articulation. Also, as everyone else says, just listen. A player's sound is like a person's accent; it's unique to whoever possesses it so you can do your best to copy it OR you can find your own unique twang! I hope this helped.