Modeling charge transport along molecular wires immersed in polarizable environments poses a grand challenge due to the high dimensionality of the problem and the various time scales involved. A previous multiscale nonequilibrium Green's function simulation scheme (Popescu, B.; Woiczikowski, P. B.; Elstner, M.; Kleinekathöfer, U. Phys. Rev. Lett. (2012), 109, 176802) has been extended significantly so that the present approach provides a much more complete physical description of the process. While the previous scheme involved the environmental fluctuations and their influence of the electronic structure of the wire, several previously neglected effects were added to the formalism: the electric field between the leads, the polarization of the dielectric environment in response to the charge present on the wire, and the relaxation of the electronic structure of the wire. Still, the underlying Hamiltonian of the wire is evaluated with electronic structure calculations, and the dynamics of the molecular system are described using molecular dynamics simulation so that (i) the formalism remains free of any model parameters and (ii) no assumptions on the underlying transport mechanism are being made. All the newly introduced details prove to affect the charge transfer along the wire markedly, while interestingly, their effects compensate each other partially. The new method is suitable for application to charge transport in junctions composed of well-defined molecular fragments, which is the case, e.g., in typical organic electronics materials. In this work, the method has been applied to hole transport through a double-stranded DNA, which nicely displays the influence of all of the newly introduced effects.