In this work, we explored theoretically the transport of organelles driven along microtubules by molecular motors of opposed polarities using a stochastic model that considers a Langevin dynamics for the cargo, independent cargo-motor linkers and stepping motion for the motors. It has been recently proposed that the stiffness of the motor plays an important role when multiple motors collectively transport a cargo. Therefore, we considered in our model the recently reported values for the stiffness of the cargo-motor linker determined in living cells (∼0.01 pN/nm,) which is significantly lower than the motor stiffness obtained in in vitro assays and used in previous studies. Our model could reproduce the multimodal velocity distributions and typical trajectory characteristics including the properties of the reversions in the overall direction of motion observed during melanosome transport along microtubules in Xenopus laevis melanophores. Moreover, we explored the contribution of the different motility states of the cargo-motor system to the different modes of the velocity distributions and could identify the microscopic mechanisms of transport leading to trajectories compatible with those observed in living cells. Finally, by changing the attachment and detachment rates, the model could reproduce the different velocity distributions observed during melanosome transport along microtubules in Xenopus laevis melanophores stimulated for aggregation and dispersion. Our analysis suggests that active tug-of-war processes with loose mechanical coupling can account for several aspects of cargo transport along microtubules in living cells.