We study the interplay of two distinct non-Hermitian parameters: directional coupling and on-site gain and loss, together with topology, in coupled one-dimensional non-Hermitian Su-Schrieffer-Heeger (SSH) chains. The SSH model represents one of the simplest two-band models that features boundary-localized topological modes. Our study shows how the hybridization between two topological modes can lead to a striking spectral feature of non-Hermitian systems, namely, exceptional points (EPs). We reveal the existence of an EP as a singularity in the parameter space of non-Hermitian couplings carrying a half-integer topological charge. We also demonstrate two different localization behaviors observed in the bulk and hybridized modes. While the bulk states and individual topological modes remain localized at the boundaries due to the skin effect, the competition between the constituent non-Hermitian parameters can overcome the strength of the skin effect and lead to the complete delocalization of these hybridized modes. We obtain explicit analytic solutions for the eigenfunctions and the eigenenergies of the hybridized modes, which exactly match the numerical results and successfully reveal the underlying cause of delocalization and the emergence of EPs.