How does Dasein co-emerge alongside its awareness/being-towards-death?

My horizon of death renders my life intelligible -implicit in all my moments-, as it makes me aware of my finitude, prompting existential anxiety, thereby creating an existential sense of urgency and worth and triggering the question of my being and time for me (none of this has to be self-reflective, explicit, of course). That is, "I project myself against the horizon of death which makes me (lead a life) possible in the first place." But doesn't this primordial projection of mine assume me in the first place? That is, how does Heidegger account for the simultaneous facts of the existence of Dasein as itself and its existence towards death?