We have all been talking only of Valmiki and not the other versions of Uttarkhand.
Adhyatma Ramayana has Uttar Kand. Sita's abandonment in the Adhyatma Ramayan:
The Devas come to Sita and tell her that it is time for the avataar to end. Rama comes to know of this conversation and decides to create the undesirable talk. He then tells her that he will leave her in the forest and she will beget 2 sons there.
Also, in the Adhyatma Ramayan, the childhood of Ramji is described as being similar to Krishna where he breaks pots and does stuff like that. Remember Kusa breaking the pots yesterday? Now, we know where the breaking pots idea came to the Sagars.