Cookie also went through a physical transformation after the first two seasons. The original look was in Boober Gorg's icon (see above
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There may have been more than one version of the puppet used during those years, but basically he had the design of the slender head. His face was more flexible as well, better enabling him to physically show anger, disappointment, etc.
I think someone wrote on this board years ago that in the third season they used parts of an old basketball to make the mouth for the Cookie Monster head we all know today - making him look closer to say Mushmellon.