The solutions of the Schrödinger with more general exponential screened coulomb (MGESC), Yukawa potential (YP) and the sum of the mixed potential (MGESCY) have been presented using the Parametric Nikiforov-Uvarov Method (pNUM). The bound state energy eigenvalues and the corresponding un-normalized eigenfunctions expressed in terms of hypergeometric functions were obtained. Some derived equations were used to calculate numerical values for MGESC, YP, and MGESCY potentials for diatomic molecules with different screening parameters (α) for l = 0 and l = 1 state with V0 = 2.75 MeV and V1 = 2.075 MeV. We observed an increase in l value; the particles behave more repulsive than attractive. The numerical values for different l-states at different screening parameters for CO molecules (r = 1.21282) and NO molecule (r = 1.1508) were obtained using the bound state energy eigenvalue of the Schrodinger equation for MGESC, YP and MGESCY potentials. Potential variation with intermolecular distance (r) for some of the particles moving under the influence of MGESC, Yukawa and the mixed potential (MGESCY) were also studied. We also observed the variation of the MGESC potential with the radial distance of separation between the interacting particles (r) for different screening parameters (α) with V0 = 2.75 MeV at l = 0 and l = 1 and YP with V1 = 2.075 MeV at l = 0 and l = 1 as purely diatomic particles in nature. The energies plotted against the principal quantum number n for different values of (α) for both CO and NO show closed resemblance even at different values of the potential depth. The energy plots of the YP and MGESC potential for both CO and NO molecules as n→∞, and the energy E→0, shows exothermal behaviour. The energy expression for the mixed potentials V0 = 5 MeV and V1 = 10 MeV, shows that both diatomic molecules possesses similar behaviour.