An interesting question, that has inspired much thought in my quater.
To my mind and ear; Sense refers to that which we intuit, implying a deeper context, one other than that of physical distance; The closeness of friendship, or other, relates to the intimacy of human contact. Where as meaning; when we consider of the etymological root; referringrefers to the physical distance, a "mean" value and, that of an average. Thus a meaning is the most commoncommonly persevered assumption, and a means is thought of as being a standard way; A means to an end. Considered then in this light; we can see and even feel where all the intimacy of sense has gone.
As suchIt is also worthy of note that the OP may also have been speaking of the nature of the relationship that had ended; this is not very clear in the information with which we are provided. The way in which the entire question is framed and understood changes; This displays to us the difference between sense and meaning, within the very context of the question.
Never the less, I conclude that the OP is making the emotional aspect stand out; Thus Sense is better than meaning, due to the nature of the situation at hand; implyingIn which She implies an inscrutable distances formfrom her shoulder,shoulder; upon which to cryshe would have cried in the past.
Those who are less emotive, may well interpret the post differently ...