There exists singular Fano varieties of dimension $n$ with a non-zero section of $\Omega^{n-1}_X\otimes \mathcal L^*$ for an ample $\mathcal L$. The existence of these examples is established in Kollár’s paper Non-rational hypersurfaces. It is unclear to me if in any of his examples $X$ is actually smooth.
Jorge Vitório Pereira
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