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I like your use of the words 'it seems', which suggests you've gained a particular degree of understanding. It does indeed 'seem' this way, but it is a very convincing mirage; an illusive trick created by the self-reflective nature of the six-sense consciousnesses.
Initially, (prior to stream-entry) one observes there to exist an enduring sense of self - personal space, time and consciousness. The realisation of stream-entry shatters this illusion. What then remains are habitual and episodic instances that Theravada calls 'I am' conceits. These 'I am' conceits are a notoriously stubborn mental fetter.
In Theravada, it is understood that one finds a way to observe the rising and disbanding of these 'I am' conceits using the five aggregate model, but it must be a method of observing whereby one is absent of any preoccupation with the themes and details of those conceits.
By doing the non-doing of this, one comes to understand anicca (impermanence) quite naturally and effortlessly simply because the transient nature of conditionality is a permeating truth. This eventually puts an end to the 'I am' conceit who's job it is to work against the tide of impermanence thus causing a strenuous relationship that we know as this and that. Anatta might then be understood as this and that being no longer defined as neither of those, nor nothing in-between them.
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First, let us notice the worry itself about the I. Once the worry is accepted as a legit concern, without even noticing, we have created drama, that will sustain the search and more things to do/achieve. The short answer is: there is no I, so the concern of making it "not be" also does not exist.
However, the short answer does not help in the marketplace where illusion is king. So we dive deeper... The I either is interested in finding out what is true (even if it means its own demise) or it is interested in deepening the illusion, (Morpheus asking Neo in "Matrix", which pill will he take) where it remains your personal God. This decision is not yours on the conscious level. The ego is always a step ahead, even in forming your preception, so there are clever strategies to disrupt the subconscious, creating a gap, for the conscious mind to get a wedge in there and from then systematically widening that gap. To see an excellent example how this gap appears, watch the movie "Revolver" from Guy Ritchie, or at least the "elevator scene" on youtube, where the main character realizes the separation from his Ego. The Eightfold Path is a good starter for strategizing.