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The intrinsically superconducting Dirac semimetal 2M-WS_{2} is a promising candidate for realizing proximity-induced topological superconductivity in its protected surface states. A precise characterization of the bulk superconducting state is essential to understand the nature of surface superconductivity in the system. Here, we report a detailed experimental study of the temperature-dependent London penetration depth, λ(T), the upper critical field, H_{c2}(T), and the effects of nonmagnetic disorder on these quantities, as well as on the superconducting transition temperature T_{c} in single crystals of 2M-WS_{2}. We observe a power-law variation of λ(T)∝T^{3} at temperatures below 0.35T_{c}. Nonmagnetic pointlike disorder induced by 2.5 MeV electron irradiation at various doses results in a significant suppression of T_{c}. These observations are markedly different from expectations for a fully gapped isotropic s-wave superconductor. Together with the substantial increase of slope, dH_{c2}/dT|_{T=T_{c}}, with increasing disorder, our results suggest a strongly anisotropic s^{++} multiband superconducting state. These results have direct consequences for the expected proximity-induced superconductivity of the topological surface states.