General movement:
* Rock: To move gently back and forth.
* Swing: To move back and forth in a pendulum-like motion.
* Teeter: To move unsteadily from side to side.
* Waver: To move back and forth in an unsteady way.
* Oscillate: To move back and forth regularly.
Influence or persuasion:
* Influence: To have an effect on someone's opinion or behavior.
* Persuade: To convince someone to do something.
* Convince: To make someone believe something.
* Move: To cause someone to feel something or to take action.
* Impress: To make a strong impression on someone.
Other meanings:
* Lean: To tilt or incline.
* List: To tilt to one side.
* Sag: To sink or droop.
* Dip: To lower or bend downwards.
To choose the best synonym, consider the specific situation and the intended meaning of "sway."