Ṣabābah is not casual affection. It signifies deep yearning and the restlessness that accompanies it. In the verse, it explains why tears flow and links an inner state to a visible bodily response.
Grammatically, ṣabābatan is accusative, functioning in the sense of mafʿūl li-ajlih, an expression of cause. It is not the grammatical subject: the tears are what overflow, while longing gives the reason.
Thus fa-fāḍat dumūʿu al-ʿayni minnī ṣabābatan means “my tears overflowed because of passionate longing,” not “longing poured out the tears” as a subject.