Others here have taken the pragmatic approach, presumably because you framed the problem of identity as having a real world affect on your daily life, arguing that regardless of your philosophical worries, you ought to act as if they are not the case. My guess is that these answers are going to be unsatisfying, seeing as your reasons for acting in such and such a way steam from your beliefs. Even if you did follow the prescriptions laid out here by the pragmatists, your worry would constantly fester in the back of your head, undermining you whenever there is space to do so. If this is the case, it would seem like we need a different approach. We need to deal with your philosophical worry. I won't presume that I can "solve the problem of identity" here, but I might be able to problematize your view, and perhaps point you towards relevant philosophical literature.

The first and most immediate problem I see is that your view is terribly (and unpalatably, imo) reductive, assuming that what constitutes who you are can be reduced to the individual atoms that you are comprised of in any given moment. What justification do you have for this? Why think that if parts of you change, you are not the same person? Why do you think you are merely your parts? etc. so on and so forth.

Also at first glance I noticed that your problem is very similar to, or at least connected to, other epistemic worries regarding skepticism--e.g. how do I know I am not a brain in a vat? There have been several ways that people have tried to address these concerns? Have you ever looked into the relevant epistemological literature to see how it might bear on your specific concerns? What about the metaphysical literature of identity? Tell me, can you give me a clear idea of what you take "being you" is, or what an continuous indent is?

That should be good enough to start with.