Based on generalization and extension of our previous work [Phys. Rev. Lett. 112, 067201 (2014)] to multiple independent Markovian baths we will compute the charge and spin current statistics of the open Hubbard model with weak system-bath coupling up to next-to-leading order in the coupling parameter. Only the next-to-leading and higher orders depend on the Hubbard interaction parameter. The physical results are related to those for the XXZ model in the analogous setup implying a certain universality, which potentially holds in this class of nonequilibrium models.