I think you are exactly right, we do have to clarify this for ourselves. Don't we replace all the cells in our body every tens years or so? Maybe not our brain cells. And some cells have a much higher turnaround, the lining of our lungs and digestive tract are constantly being sloughed off and replaced, I think. But we nevertheless carry with us the feeling of mental and physical continuity.
The individual qua individual can't find a "me" or an "I". We have to credit Hegel with this insight. The self-definition of the self as self comes when we encounter the other, and the object in general. We have to come against limits to begin to define ourselves.
Then the question is whether the "Bildungsroman" of lived life is enough. This Bildungsroman-process is our education, and the hard knocks we take in life, the education of Spirit/Mind. But in addition to this it seems we have to cast ourselves forward, we need an intention toward the future to maintain the "me"; this we see really in Nietzsche, and the Heidegger of Being and Time, and in Sartre too. The background here is Schopenhauer, and maybe Kierkegaard too, I am no so familiar with Kierkegaard.
But it's complicated in that we tend to fool ourselves into thinking we are unique when in fact we are not. We are captured by the "They", Heidegger, or by our instincts, Freud, everything we think WE are is merely the product of total mediation (Frankfurt School) It's not really our thought, it's something we heard on the news or some such.
When we say "me" or "I" this is a self-reification. And as Adorno said, in all reifications there is a forgetting. We tend to forget not only the history of our life and education, but the totality of the world that is wrapped up into us.
It is sadly possible to see ourselves today merely as a reified object for sale on some market, the job market or the love and sex market. The human in us is crushed down into an object to be marketed, and ultimately to be thrown away. This sort of alienation away from the fullness of Being is kind of ceasing to be.
So now we are lost again, where is the "me"?
The existentialists are probably right that we can (we must), in an atomized world, for our own mental health, and even if it is a fiction, transcend the in-itself by some projection forward (goals, whatever). Even if it is a fiction, we hold ourselves together and organize ourselves by being in and for "ourselves".
The loss of self-organization, or the loss of the necessary fiction of the organized self or me, would be the ceasing to be, or psychosis. It's not so much the physical body, since we discharge and replace the physical body all the time. The end of the self-narrative projected forward would be the ceasing to be.
P.S. The problem today is that the pressure for the narrative is all on us. Society and religion used to provide "the" story, now under postmodernism and in a porous world, we have too many stories, and life is short. The need for the personal story would be less if there could be "our" story, but this would require a less atomized world. We can hope that existentialism is only a solution for this present time in our history.