We present a method for calculating the full counting statistics of a nonequilibrium quantum system based on the propagator noncrossing approximation (NCA). This numerically inexpensive method can provide higher-order cumulants for extended parameter regimes, rendering it attractive for a wide variety of purposes. We compare NCA results to Born-Markov quantum master equations (QME) results to show that they can access different physics, and to numerically exact inchworm quantum Monte Carlo data to assess their validity. As a demonstration of its power, the NCA method is employed to study the impact of correlations on higher-order cumulants in the nonequilibrium Anderson impurity model. The four lowest-order cumulants are examined, allowing us to establish that correlation effects have a profound influence on the underlying transport distributions. Higher-order cumulants are therefore demonstrated to be a proxy for the presence of Kondo correlations in a way that cannot be captured by simple QME methods.