You should try speaking other Accent as foreign accent.

To answer seriously
You need to change the way you think and exercise control of what comes out of your mouth.

Practice more writing as writing slows your thought down. The problem with accent is that it would make you speak before you think.

Use gestures to make others wait for your answer so you take extra time to think.

Cut out cliche statements that every one uses that will expose you as a southerner.