Thanks! The reason why "I'm sitting down" is correct is because the action "sitting" continues to happen after you put your body on the chair. Is this right?
No. We can use 'sitting down' for either the action of lowering one's body onto a chair or for the state of being seated. In the latter situation we frequently don't use 'down', but we can.
All verbs of body posture have the same pattern. They can refer to the action of taking a particular posture OR to the state of being in that posture.
ACTION .......................................... STATE
I am/was sitting (down) ......... I am/was sitting (down) / I am/was seated. I am/was lying down ...............I am/was lying down I am/was kneeling (down) ......I a