We establish sufficient conditions for the existence of moments of the steady-state queue in polling systems operating under the binomial-exhaustive policy (BEP). We assume that the server switches between the different buffers according to a pre-specified table, and that switchover times are incurred whenever the server moves from one buffer to the next. We further assume that customers arrive according to independent Poisson processes, and that the service and switchover times are independent random variables with general distributions. We then propose a simple scheme to approximate the moments, which is shown to be asymptotically exact as the switchover times grow without bound, and whose computation complexity does not grow with the order of the moment. Finally, we demonstrate that the proposed asymptotic approximation for the moments is related to the fluid limit under a large-switchover-time scaling; thus, similar approximations can be easily derived for other server-switching policies, by simply identifying the fluid limits under those controls. Numerical examples demonstrate the effectiveness of our approximations for the moments under BEP and under other policies, and their increased accuracy as the switchover times increase.