The problem with this 'it died after its master died' is, that the Watcher probably was not in league with Sauron, let alone mastered by him. the Watcher was one of the nameless things from the deeps of the earth, older than Sauron and unknown to him, as Gandalf says somewhere (paraphrased). The Watcher was 'just' attracted to the One Ring, because it was a source of evil and power, just as it itself was. It wanted to have this source of power, it was not ordered by Sauron to take it for him.
At least, this is what I think about it
