I find the question a little awkward but important.
In metaphysics 'making sense' would mean being logically coherent. So, where the existence of a thing would not be logically coherent we would assume its non-existence. The usual measure is contradiction, such that if the existence of a thing would cause a formal contradiction then we would assume it does not exist.
The complication is that our usual idea of Existence doesn't make sense, suggesting that nothing exists as we usually think it does. So, we have a choice. We can believe things exist as we usually think they do despite the incoherence of their existence, or we can assume that things do not exist as we usually think they do.
These two views coincided roughly with 'Western' thought, which is stereotypically naively-realistic, and 'mysticism' or the Perennial philosophy, for which Existence would not be what we usually think it is.
It is an odd philosophical fact that the latter is more in accord with reason and logic than the latter. Thus Russell, having rejected the latter view, must dismiss metaphysics as useless, while the 'mystical' view he works so hard to avoid makes full use of it.
Your p and q argument is fairly meaningless unless you carefully define what you mean by 'make sense' and 'existence', but it might work with the right definitions.
Perhaps you could ask yourself whether you can make sense of Existence. Your argument suggests that if you cannot do so then nothing exists. This seems a rather strange argument. I would rather say that if you cannot make sense of existence within a metaphysical theory, its origin and nature, then things do not exist in the way the theory states.
Thus rather than say a thing cannot exist if its existence would be logically incoherent, (such that its existence would be a 'true contradiction'), it would be better to say either it does not exist or our idea of 'things' and their existence is incoherent. Which it is would be for us to figure out as a separate issue.
If your idea of existence is incoherent then your argument will force you to deny the existence of everything. If your idea of existence is coherent then I think it might work, but you'd have to clarify what you mean by 'make sense'.
EDIT: The question title seems to have changed, or maybe I misread it. So, I'll just add that there is no evidence to suggest the world does not make sense, but plenty to suggest it is not easy to make sense of it.