First of all, a simple analogy that will (hopefully) make you understand the question you're making:
Self-awareness here is the water: it's the embodiment of life (lucidity) itself and there's no replacement for it. Every food (read technique) that helps you achieve lucidity contains a small portion of water (self-awareness).
ADA is the meat: it's simply one type of fuel in order to reach lucidity (self-awareness), but it's not the only kind there is, nor is it crucial to achieve lucidity.
Assuming you understood the analogy above, you realize that this is not than a question of should you practice one or another. And this brings us to the next principle:
Water is not food: it's a nutrient. (or for those who are saying "zoth, shut up with the analogy!" which will upset a certain TED talk guy who taught zoth that metaphors are essential to human comprehension) -
Self-awareness is not a technique: it's a state of consciousness
States of consciousness are states with unique brain activity. They are still not completely understood, but we know that they probably regulate aspects like the way information is processed, perception, motor output, sleep, etc. Now, everything that doesn't refer to our normal state of consciousness (consisting on a range of frequency that we call beta waves) we call "Alternate states of consciousness": scenarios of ASC can refer to dreams, drugs, sensory deprivation, certain behavior (like yoga, meditation*), or even hypnosis. In these scenarios, the aspects mentioned above can be altered, essentially making the brain react and respond differently to stimulus, or experiences changes on neuronal activity.
In that sense, the self-awareness specific to the state of dreams is what we call lucidity: lucidity in itself is a state of consciousness that is pointed out as resembling a state of consciousness like the waking life and at the same time with the ASC of meditation. Lucidity is what we call to a certain altered state of consciousness in a dream in which brain activity changes, particularly in regions responsible for self-monitoring: the individual becomes able to assess their own state ("I am dreaming"), to conscious plan and act according to his will, and more importantly, to distance himself from the experience. If we had to give lucidity a name for our waking life state of consciousness, that name would be (drums drums drums)...self-awareness
This is also why we perceive lucid dreams as a middle point between 2 spectrums of consciousness: dream and waking life. When an individual starts achieving this specific brain frequency in a dream, he reaches the state of self-awareness or what we call lucidity, and stepping in the edges of this state has pretty visible consequences:
- Too low introspection and you risk drifting into the altered state of consciousness that we call (non-lucid) dreams;
- Too high introspection and you risk drifting into the altered state of consciousness that we call waking life.
(Time to start answering your question I guess, you probably aren't interested on this wall of text

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ADA by itself doesn't put you in lucidity state: it's not an altered state of consciousness, it's a technique designed to improve the awareness of your surroundings. In that sense, it may help you with other aspects that might be useful for lucid dreaming: put you more alert for possible lucidity cues (like dream signs, or anomalies with the dream), make you more focused and more prone to engage in certain habitual actions (like doing reality checks), etc. Like the creator of the technique itself tells you in his tutorial, you should be making reality check and wondering if you're dreaming while you pay attention to the reality around you.
So to really answer your question, is not a matter of practicing self-awareness or ADA, but more like: is ADA the way to go? The answer is: ADA is a pretty good technique yes, but don't forget to add certain behaviors that put you towards a state of self-awareness during your practice (basic behaviors being reality checks, analyzing your thoughts, perceiving yourself as separate of the experience you call reality). Good luck!
* This why is never too much to stress the importance of understanding what meditation you are performing, as one type of meditation is not necessarily beneficial (or at least, influential) towards a state of self-awareness.
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