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In this paper, we studied the quantum dynamics of the TiC, NiC, and CuLi diatomic molecules interacting with the Varshni-Eckart potential (VEP) model in topological defects geometry associated with cosmic string under the influence of magnetic and Aharonov-Bohm (AB) flux fields. We employ the Greene-Aldrich approximation scheme in the centrifugal term and determine the approximate eigenvalue solution using the Nikiforov-Uvarov-Functional Analysis (NUFA) method. The numerical energy spectra for topological defects geometry and external fields with this potential are determined. Subsequently, we applied the eigenvalue solution to various cases of the potential model and obtained their analytical solutions. When compared to Minkowski flat space results, the presence of topological defects and external fields eliminates degeneracy and shifts the energy spectra of the diatomic molecules. [ABSTRACT FROM AUTHOR]