To describe the elastic scattering cross sections of C12−12C and O16−16O systems, in our recent work (RM1), the double-folding (DF) model potentials was calculated via the density-dependent averaged effective two-body nucleon-nucleon (NN) interaction (DDAEI). The DDAEI can be generated through the lowest order constrained variational (LOCV) method for the symmetric nuclear matter (SNM), with the input bare Reid68 NN potential. Then the DF potentials were directly inserted into the FRESCO code to find the differential cross-sections (DCS). The result was encouraging with respect to the experimental data and the other theoretical works. In the present report, the above folded potentials are fitted through a Woods-Saxon (WS) shell model potential, which can easily be used (through the WS parameters) as an input to the FRESCO code, without any approximation. The calculated DCS are compared with the corresponding calculations coming from the fitting procedures with the input finite range DDM3Y1-Reid68 potential, RM1 and the available experimental data at different incident energies. As in RM1, it is shown that a reasonable description of scattering data at the low and the medium energies for the elastic scattering of the C12−12C and O16−16O systems can be obtained in the framework of the above LOCV-DDAEI, without any need for defining a parameterized density dependent function in the effective NN potential, which is formally considered in the typical DDM3Y1-Reid68 interactions. The χ2 fits are calculated and discussed. Our result based on DF WS potential parameters, i.e., WS-FRESCO DCS, are approximately in agreement to those of RM1. Finally, the generated energy dependent WS potentials can be directly used in the future calculations and considered as an approximate energy dependent shell model one-body potential which is based on the phenomenological NN interaction.