This paper investigates the adaptive state-feedback stabilization problem for stochastic nonlinear systems. Compared with the existing results, we consider more general and more practical systems, i.e., systems with time-varying powers and unknown covariance simultaneously. We propose a new adaptive state-feedback control method, which ensures that the closed-loop system is globally stable in probability and the states are regulated to the origin almost surely. Finally, the feasibility of the design method is verified by two examples.