It is true that a being, independent of the body, lives through a body. At the moment of death, when life departs from the body, there is no change in the apparent physiology of the body. Never the less, the dead body immediately starts to decay and to putrefy emitting foul smell. Devoid of life (life in its essence is an integral part of the Soul till it attains salvation), the body acquires an abhorable state. It proves that although the being enlivens the body, it has an existence of its own, independent of the body. In parlance of spirituality, this separate entity is called atma (Soul). This Soul is our true identity. Your right answer to the question 'Who am I?' is 'I am That Atma'.
Say the Upanishads:-
"'The Soul cannot be perceived by sermons, or through intellectuality, or by too much listening to discourses on spirituality."