Sentences like 'The more we eat, the less we want' or 'The more you read, the more you know' are often heard, but is it grammatical to say sentences like 'The more we eat, the more we don't want'; that is, negating the clause that follows the comparative phrase?
It's more usual to use 'the less' in the 2nd clause, where you want to express a negative:
1. The more I eat, the less I want.
2. The more I hear, the less I listen.
Your form ("the more I eat, the more I don't want") could be used for emphasis, though. Since it's less usual, it would have an element of surprise.