I'm familiar only with one type of False awakenings. They are a normal, non-lucid dream, in which you wake up and go about your business. Same way, as you do in your waking life.

Noticing your dreams could be a sign that yes, indeed, your mind does pay more attention to them, since you are thinking about dreaming more. So I would say it's a good sign.

If you need any help, just let us know. happy dreams

Induction Methods and Techniques