How to completely be aware of each moment? How can I be completely aware in each and every moment, even working, walking, or talking? Is "continuous breath awareness" is an answer?
You can still be aware of the breath. In case of working it is more easy to see the expansion and contraction nature / sensation of breathing process. In which case it might be an idea to look at your chest or abdomen for rising and falling sensation.

When you walk, talk or do anything else you need mindfulness to do the task right. In Buddhism this awareness is not too tied to being aware of what you do. What needs to be done is to be aware of the arising and passing of sensation (phenomena what is felt) with equanimity. In the case of breath any sensation pertaining to the breath like expansion contraction or touch at the upper lip or nose or length will be sufficient.