Difficult questions, and great ones to think about!
There are multiple possible ways to think about awareness and self, and they produce different types of experiences. If you want to explore it, one classic way is to ask, when I say "I know that I am" does that statement describe my body ("there is a body and I know that there is a body") or does it describe knowing ("there is knowing and I know that there is knowing") or does it describe perception, or feeling, or something else? Of course, we tend to bundle all those things together, but if we look closely, they actually come and go in awareness, one at a time, or two or more at close to the same time, or whatever, but we can certainly say that not all of the possible components that make up "I" are present in awareness all of the time. So, instead of "self-awareness" we might want to say "body-awareness," "knowing-awareness," etc. Now if we look at the same question of what happens during sleep, or even when suddenly switching from one situation to the next while awake, there is an interesting question of whether there is just one awareness, with multiple things arising in it, or whether awareness itself arrises afresh each time. There are some advantages of each way of thinking about it, but I like the second one.
Rob Burbea wrote a great book on this subject called Seeing That Frees.
