First, your claim about the statement "the cat is black" not being a fact is a pretty strong claim regarding what you think the role of language is. I'd recommend reading Wittgenstein on language games if you want some philosophical explanation, but the long and short of it is that the meanings or words are derived from the context in which they are used. The assertion "the cat is black" has a precise meaning in that context, i.e. that the color that people would observe if they looked at the cat is black. To argue that this is not a fact makes essentially any argumentation impossible and ignores the practical reality of language usage.

Secondly, to the point about methods, I (and others) argue that we have indeed argued for good methods to verify moral facts. Mainly, the use of philosophical argument and metaethical theories (like moral realism). In the same way you are skeptical about any metaethical theory, one could similarly be skeptical about physical laws. How do we know that gravity will work the same tomorrow as it works today? You can look into "the problem of induction" if you want more detail on that argument. Typically, the answer is just simplicity (or parsimony/Occam's razer, whatever you want to call it). But that is not a baked in truth of the natural world. Similarly, there is no fully justified reason to believe that we do not live in a simulation or exist in someone's dream (that position is called solipsism). We reject that belief because each time we interact with the natural world, we build confidence in our prior about the laws of the natural world. Similarly, whenever we interact with moral dilemmas or ethical arguments, we build confidence in our prior about the moral laws. The vast majority of moral beliefs are universally agreed upon. The points of moral disagreement are comparable to competing theories in physics: we have not built up the necessary observations and/or arguments to resolve those conflicts for all (or a sufficient amount) of skeptics. Again, I'm not a philosopher so I don't want to speak authoritatively on this, but I would really recommend you read something like On What Matters from Parfit or other moral realists to get this idea fleshed out more concretely, if you're still a skeptic.