As emission standards become stricter, more vessels are deciding to use LNG as fuel. The primary challenge with this change is that external sloshing excitation from the ocean may cause rapid pressure drops in fuel tank, resulting in engine shutdown. In this paper, a computational fluid dynamic (CFD) model based on the volume-of-fluid (VOF) method and mesh motion treatment is constructed to predict the pressure drop and thermal response in LNG fuel tanks under sinusoidal excitation. The sinusoidal excitation and phase change model is realized by the User-Defined Function (UDF). The effectiveness of the CFD model is verified by comparing with related fluid sloshing experiments. The thermal physical processes in the LNG fuel tank under sloshing conditions and the static condition are compared, and the effects of the sloshing amplitude and frequency on the pressure and temperature distribution of the tank are also studied. The numerical results indicate that the sloshing leads to the mixing of vapor and liquid, enhances interfacial heat and mass transfer, uniforms the temperature distribution, and promotes the pressure drop of tank. In particular, the influence of sloshing excitation on heat and mass transfer is theoretically analyzed using the sloshing Nusselt number. A feasible simplified method is provided for the complex phase change problem in cryogenic tanks under sloshing conditions.