(1) Is this a question of aversion? How can I avoid being phased by negative states?
(2) More specifically, how do I avoid identifying with such negative states, and how do I recognize their impermanence when they occur?
The answer is pretty simple; be mindful and guard the sense-doors.
If one is not mindful of an object impinging on the sense-doors, craving will arise, i.e. liking or disliking. If one is not mindful, insight into the Three Marks of Existence, cannot occur.
Here is a quote by Ven. Mahasi Sayadaw on exactly this topic;
"If one does not observe mental and physical phenomena every time they arise at the six sense doors, one cannot realize that there is nothing to them but mind and body, which are conditioned, impermanent, unsatisfactory, and not-self. As a result one will develop an attachment to the objects that one fails to observe.
If, on the other hand, one observes mental and physical phenomena the moment they occur, one will realize that there is nothing to them but mind and body, which are conditioned, impermanent, unsatisfactory, and not-self.
As a result one will be free from attachment to objects that one is able to observe. Thus the wholesome (kusala) action of insight liberates one from attachment and thus is considered renunciation. This renunciation, in turn, liberates noble ones from the cycle of suffering (saṃsāra) by developing insight step by step until nibbāna is attained. And so the wholesome act of insight is called “the great liberation of noble ones” (ariyanaṃ niyyanaṃ) because of its liberating effect".
-- Manual of Insight, p. 58, by Ven. Mahasi Sayadaw.