A Nihilist is more stable than you think. A tree doesn’t need purpose to grow, and neither does a human. Instead they follow a process that’s expressed by the body in response to the environment. For humans we not only have physical sensation but also memories, ideas, and thoughts. Without a worldview these all still happen, and your body will still react to your thoughts and sensations without a purpose. A healthy Nihilist would not feel sad about having no meaning, since they never had one anyway; on reflection, they might feel happy that there’s no obligation to respond other than how their situation moves them to. A Nihilist can do what is morally right without the restrictions of a purpose that proscribes them to do otherwise.
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Longer take:
Defining what is morally right requires value judgement, and Nihilism rejects morality. What Is (facts) can't tell us what we ought to do. [1]
To build a moral framework on what one ought to do, you must necessarily ascribe meaning and value judgement otherwise it can't claim to be moral and it would merely be actions you perform devoid of meaning.
[1] https://en.m.wikipedia.org/wiki/Is%E2%80%93ought_problem
Why do you think meaning is required for making a value judgement? I can very easily claim that killing humans is wrong without relying on any metaphysics, just like I can claim the sky is blue. Both are beliefs based on reasons; meaning can be discarded.
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Morals/meaning have a shaky absolute foundation, in that deep down nothing probably matters and the universe is indifferent.
However I'm a person, born in modern society; everything has relative meaning even if the epistemological foundations are shaky.
I see Nihilism similar the mathematicians trying to prove foundational math [1], and similarly doubting it's actually fundamental.
Yet here we are using math to amazing things! Math doesn't need to be true for it to be useful. I find useful things valuable.
Meaning, in particular shared meaning, is similarly valuable and worth pursuing even if they can't be proven.
I agree but I think the breakdown is even stronger than that: Nihilism gives you no way to decide what is "better" in the first place.
If you've gotten to Nihilism, you have a blank slate to build what you Do believe.
Ascribing meaning to things has value. For example, human life in Nihilism is unimportant, which may or may not be true, but we all benefit from the shared belief in the value of life even if it's technically wrong.
So don't stop at Nihilism as the conclusion but rather use it as a canvas and build up what's meaningful to you. With any luck, others will find the same things as you meaningful.
It was my way out of Nihilism. The pursuit of being good is more important and valuable than us all agreeing on what is good (or even that good exists at all). And given that goal, all you can do is try your best.
The trouble revolves around how much judgement is required on any given subject. On the right subject people can come perilously close to laying out facts devoid of interpretation; on the wrong subject either no one can or no one believes anyone can - which is pretty much the same thing.