Jaʿaltu is a past verb with a first-person agent. In ʿiṣāmahā, ʿiṣām is the object and the feminine pronoun hā refers back to the water-skin. ʿAlā kāhilin minnī locates the carrying strap upon the speaker's own shoulder.
Dhalūlin muraḥḥal qualifies the shoulder or load-bearing person as accustomed to service and travel. Minnī makes the physical experience unmistakably personal, while qirbati aqwāmin shows that the object belongs to others.
There is thus a relation between the poet's self and other people: after the private suffering of the night, he now bears a communal load. The line expands his characterization rather than merely adding another travel detail.