I am looking for a word (or two words) that means something similar to "unagi" as it is used in the TV show Friends (i.e., the concept they were trying to go for, not the fish!) - a total sense of awareness of yourself and your surroundings.
In Friends, Ross uses the word unagi incorrectly, but the word he means to use is zanshin, which is a term used in Japanese martial arts.
Zanshin means "residual mind". It refers to a state of relaxed awareness in which a practitioner of martial arts is wary of their surroundings before, during, and after attacks. It can refer to mental states, physical states, or both, but the basic idea is that the user is always aware of their surroundings, even when they are at rest. So if I were to give you a word (or two) to summarize the concept you're thinking about, it would be relaxed awareness.
The term "ecstasy" comes from a Greek root meaning "to stand outside". It is a subjective experience of transcendence, during which the individual achieves a heightened sense of awareness.
We talk about the need for living in the now, about mindfulness (focusing one's awareness on the present moment, calmly acknowledging and accepting one's feelings, thoughts, bodily sensations, and awareness of surroundings.)
According to Jon Kabat-Zinn:
My own definition of mindfulness is very simple: Mindfulness is the gentle effort to be continuously present with.
Assuming it is zanshin you mean, I suggest "poise" especially senses 4 and 5 in wiktionary: "A state of balance, equilibrium or stability; composure...".
See also the OED definition "Balance, equilibrium... composure... self-possession" (subscription link; see sense 7, especially 7(b)).
In particular, "poise" seems to have the same connotation of readiness. Note also some of its synonyms: calmness, serenity, self-possession, presence of mind, balance, bearing.
It is not a perfect equivalent, because "poise" has a few other meanings that do not match so well.
Situational awareness describes this perfectly.
Situation awareness (SA) involves being aware of what is happening in the vicinity, in order to understand how information, events, and one's own actions will impact goals and objectives, both immediately and in the near future. One with an adept sense of situation awareness generally has a high degree of knowledge with respect to inputs and outputs of a system, i.e. an innate "feel" for situations, people, and events that play out due to variables the subject can control.
While this term has been co-opted into business jargon, it originated in military theory. Elements of it were featured prominently in Sun Tzu's Art of War.
In the business sense it has come be associated with adapting toward achieving one's goals. But, its original sense was being able to function as a solider due to the awareness of one's surroundings.