The fully nonlinear Chalikov-Sheinin (CS) model, based on a nonstationary conformal surface following coordinate transformation that reduces the principal equations of potential waves into two simple evolutionary equations, is firstly used to study the dynamics of nonlinear gravity waves, mainly involving the propagation of Stokes wave with disturbed sidebands, the evolution of one wave packet and the interaction of two wave groups. The numerical results are compared with those obtained by the HOS method, which is an approximately fully nonlinear scheme if the order of nonlinearity retained is not large enough. In most cases the results are consistent between these two numerical models. However, the fully nonlinear CS model can maintain a higher accuracy and better convergence in the long-term evolution process. Statistical properties of mechanically generated unidirectional nonlinear irregular waves are also simulated with the CS model and compared with numerical results derived by the temporal version of MNLS equation. Except for the wave height distribution, CS model shows a little better performance in reproducing the wave statistics, especially in term of fourth-order normalized cumulants and the intermediate probability of crest and trough exceedance distributions under the condition of strong nonlinearity.