I think that while the knife in the dark did affect Frodo, he would have still given in at the end. It was partly just the amount of time he spent with the ring. Another part was that he felt the dark part of the ring and Sauron more when he was near mount doom. Frodo had already had the ring for a long time, and there was no way to get rid of it without going into the middle of Mordor.
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"Wake up! Wake up! Wake up, sleepies, we must go, yes, we must go at once."
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