Dielectronic recombination (DR) of singly charged ions is a reaction pathway that is commonly neglected in chemical models of molecular clouds. In this study we include state-of-the-art DR data for He$^+$, C$^+$, N$^+$, O$^+$, Na$^+$, and Mg$^+$ in chemical models used to simulate dense molecular clouds, protostars, and diffuse molecular clouds. We also update the radiative recombination (RR) rate coefficients for H$^+$, He$^+$, C$^+$, N$^+$, O$^+$, Na$^+$, and Mg$^+$ to the current state-of-the-art values. The new RR data has little effect on the models. However, the inclusion of DR results in significant differences in gas-grain models of dense, cold molecular clouds for the evolution of a number of surface and gas-phase species. We find differences of a factor of 2 in the abundance for 74 of the 655 species at times of $10^4$--$10^6$ years in this model when we include DR. Of these 74 species, 16 have at least a factor of 10 difference in abundance. We find the largest differences for species formed on the surface of dust grains. These differences are due primarily to the addition of C$^+$ DR, which increases the neutral C abundance, thereby enhancing the accretion of C onto dust. These results may be important for the warm-up phase of molecular clouds when surface species are desorbed into the gas phase. We also note that no reliable state-of-the-art RR or DR data exist for Si$^+$, P$^+$, S$^+$, Cl$^+$, and Fe$^+$. Modern calculations for these ions are needed to better constrain molecular cloud models.