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The compounds CrX₃ (X=Cl, Br, I) are layered materials that have received much attention for their few-layer magnetism and potential to host topological magnons. Two types of magnetic order are observed: out-of-plane ferromagnetism (FM) for CrBr₃ and CrI₃, and in-plane FM order with the spin direction alternating antiferromagnetically in the perpendicular direction for CrCl₃. Nearest-neighbor FM coupling on the honeycomb lattice layers would result in a graphene-like magnon dispersion with Dirac nodes at the K point, though in CrBr₃ and CrI₃ gaps due to the Dzyaloshinskii-Moriya interaction have been reported. For CrCl₃, however, we report neutron scattering measurements showing that gapless Dirac magnons are present at low temperature. For higher temperatures, the magnon features gradually soften and broaden on the scale of 10s of K, with only subtle changes at T_N~14 K. On cooling below ~50 K, an in-plane negative thermal expansion and a phonon anomaly are seen, in contrast to CrBr₃ where such behaviors occur abruptly at the ordering temperature. DFT calculations show that these behaviors arise from increasing in-plane spin correlations. We suggest that differing anisotropy (easy-plane for CrCl₃ vs. easy-axis for CrBr₃) are responsible for the different temperature dependence.