Before suggesting some answers, I will note that the word body can be used in various ways. The comment by user067531 is correct: body can include the head. If someone wants to be clear that they are talking about the full body (with head), the most normal and common expression is "whole body". This corresponds to sense 1) in this dictionary definition.
My whole body sways to the rhythm.
However, you are correct that there is another sense of body, which excludes the head. This is sense 3) in the linked definition above.
The blow almost severed his head from his body.
There are many other words that mean the whole physical/biological body of a person (including the head). There are also many that refer to the whole person, including the physical body and thoughts, feelings, perceptions.
Some of these words make very subtle distinctions between different aspects of a human being. For example the word soma means
The body as distinct from the soul, mind, or psyche.
Words like physicality, corporeality and organism can be used in a similar way.
If you mean that every part of you - physical and mental - is affected by the music, then you can just use the pronouns I or me without any extra explanation.
The music moved me.
You could replace "me" in that sentence with one of these: "my being"; "my whole self".
There are also various philosophical and scientific questions about how mind and body are connected, and the difference between the subjective experience of one's body and the objective experience of someone else's body (or your own body image), and so on... but those go beyond the remit of this website!