Reactions of mode-selectively excited HOD+ with C2H2 and C2D2 were studied over the center-of-mass collision energy (E-col) range from 0.15 to 2.9 eV. HOD+ was prepared in each of its fundamental vibrational states: ground state (000), bend (010), OD stretch (100), and the OH stretch (001). Charge transfer is the dominant reaction at all energies, although it is inhibited by increasing E-col, and is accompanied by hydrogen exchange. The total charge transfer cross section is similar for C2H2 and C2D2, however, the tendency toward charge transfer with hydrogen exchange (CTHE) is significantly greater for C2D2 compared to C2H2. Charge transfer shows no significant effects of HOD+ vibrational excitation, however, CTHE is significantly enhanced by vibration at E-col < 0.62 eV. Both H+ and D+ transfer reactions (HT, and DT, respectively) are observed for both C2H2 and C2D2, with little dependence on collision energy, but with mode- and bond-specific enhancements from excitation of the OH and OD stretches. Recoil velocity measurements show that all channels are direct, except perhaps at the lowest collision energies. Mode-specific effects on the recoil velocity distributions are also observed, revealing how vibrational excitation affects reaction at different collision impact parameters.