In this paper, we provide new examples of Levi-Civita Ricci-flat Hermitian metrics on certain compact non-Kähler Calabi-Yau manifolds, including compact Hermitian Weyl-Einstein manifolds, compact locally conformal hyperKähler manifolds, certain suspensions of Brieskorn manifolds, and every generalized Hopf manifold provided by suspensions of exotic spheres. These examples generalize previous constructions on Hopf manifolds. Additionally, we also construct new examples of compact Hermitian manifolds with nonnegative first Chern class that admit constant strictly negative Riemannian scalar curvature. Further, we remark some applications of our main results in the study of the Chern-Ricci flow on compact Hermitian Weyl-Einstein manifolds. In particular, we describe the Gromov-Hausdorff limit for certain explicit finite-time collapsing solutions which generalize previous constructions on Hopf manifolds.