‘I am feeling all right’ means ‘I am all right physically’. After giving the injection the doctor may ask, “How do you feel now?” You might respond by saying ‘I am feeling all right’ or ‘I don’t feel all right, I am feeling giddy’. ‘

I am doing all right’ means ‘I am prospering, my business is flourishing’. ‘I am doing well’ means, ‘I am all right, business-wise, workwise, etc.’ It has nothing to do with your physical well-being”