For me thirst must be the absence of peace.
I would read it as fear deprives us of peace even in safety/plenty.

One of the things I like about proverbs is that we each read them through the filter of our own experience.
We use them to evaluate cognitive function - the meaning is less important than the patient's ability to think in the abstract. e.g. "a rolling stone gathers no moss" might be explained as "if you stop trying anything new you get stuck in a rut" or "if you can't stick to something you won't succeed." On the other hand the response "if you roll a rock over the moss gets scraped off" makes you worry about their ability to manage complex situations.