The modal analysis of a lossy coupled plasmonic waveguide consisting of two graphene sheets sandwiched among three SiO2 layers is presented. This analysis extracts even and odd complex propagation constants in addition to even and odd characteristic impedances at various Fermi levels. It is shown that the first even mode is lossless in theory, but other modes are very lossy. Taking into account the losses, first, the transmission line (RLCG) model, including self and mutual elements, is constructed for various Fermi levels and frequencies. Then, the length of a lossy plasmonic directional coupler based on the coupled plasmonic waveguide is designed for a variety of Fermi levels. The transmission, reflection, coupling, and isolation coefficients of the device are obtained using advanced design system for several Fermi levels and coupler lengths to maximize the power at the coupled ports and minimize it on the reflected and isolated ports. The computational results show that the minimum length of the plasmonic coupler is 39 nm at 40 THz. Sensitivity analysis shows that increasing the frequency, decreases the length of the optimum device while increasing the sheets’ Fermi level. Finally, Comsol simulations of the coupled plasmonic waveguide are utilized to verify the even and odd modes.