Escurres comes from the verb escurrir, meaning "to drip," "to drain," or "to slip away." While it can refer to liquids, it's often used metaphorically.
In "Cobarde," Dread Mar I sings, "Ver que te me escurres por los dedos" (Seeing you slip through my fingers). This vivid imagery conveys a sense of helplessness and loss, as he watches someone he cares about slowly distance themselves from him, like sand or water slipping away.
Dread Mar I opens "Cobarde" with a confession baked into the title: "Es verdad que soy / Un poco cobarde, pero vos" [It's true that I'm / a little cowardly, but you]. He isn't the one falling apart — she is the one going quiet on him, and he calls that the colder hurt: "No es sentimental / Ver que te me escurres por los dedos" [It's not sentimental / to watch you slip through my fingers]. What actually stings is her silence: "No cuidas nunca tus palabras / Es fuerte ver que no me amas / Y no dices nada" [You never mind your words / It's hard to see you don't love me / And you say nothing]. By the last verse the lyric strips down to one repeated word, "Nada" [Nothing] — the song ending exactly where the conversation did, with nothing left to say.