No, it isn't a weird interpretation because it's made with the context of when the quote is said. This is also referenced by the last line in the post. This is the context:

Then as quickly as he could he cut away the binding cords and laid his head upon Frodo’s breast and to his mouth, but no stir of life could he find, nor feel the faintest flutter of the heart. Often he chafed his master’s hands and feet, and touched his brow, but all were cold. ‘Frodo, Mr. Frodo!’ he called. Don’t leave me here alone! It’s your Sam calling. Don’t go where I can’t follow! Wake up, Mr. Frodo! O wake up, Frodo, me dear, me dear. Wake up!’ Then anger surged over him, and he ran about his master’s body in a rage, stabbing the air, and smiting the stones, and shouting challenges. Presently he came back, and bending looked at Frodo’s face, pale beneath him in the dusk. And suddenly he saw that he was in the picture that was revealed to him in the mirror of Galadriel in Lo´ rien: Frodo with a pale face lying fast asleep under a great dark cliff. Or fast asleep he had thought then. ‘He’s dead!’ he said. ‘Not asleep, dead!’ And as he said it, as if the words had set the venom to its work again, it seemed to him that the hue of the face grew livid green. And then black despair came down on him, and Sam bowed to the ground, and drew his grey hood over his head, and night came into his heart, and he knew no more.