In this work, the authors systematically investigate the strong coupling constants for vertices involving doubly charmed baryons (ẞ_cc), singly charmed sextet baryons (ẞ_c⁶), and D mesons, using the LCSRs method. The calculations are performed analytically and numerically, with the non-perturbative input taken as the light-cone distribution amplitudes of the singly charmed baryon ẞ_c⁶. Through graphical analysis, the authors determine the optimal Borel parameter ranges and present numerical results with a thorough error estimate, accounting for uncertainties in the Borel parameters, the continuum thresholds and decay constants. To extrapolate the b-baryon LCDAs to the case of c-baryons, the authors adopt a simple scaling method, and the uncertainty from the scaling factor is incorporated to reflect this extrapolation. The uncertainties induced by the decay constants f_ẞcc and fẞc⁶ dominate over those from other parameters, which can be attributed to the larger inherent uncertainties of these input parameters. The LCSRs results deviate from the ratios predicted by SU(3) flavor symmetry, which is primarily due to the significant differences in the light-cone wave functions of baryons with different light quarks (u, d, s) in the LCSRs calculation. Furthermore, these deviations can be well accounted for by the ratios of input parameters such as masses and decay constants. The results can be cross-checked by other approaches such as conventional QCD sum rules or lattice QCD. These strong coupling constants are key inputs for analyzing non-leptonic weak decays of doubly heavy baryons, particularly for evaluating final-state interactions in charmed-bottom baryon decays. Our findings thus provide both theoretical insights into the strong and weak decay dynamics of doubly heavy baryons and a practical tool for experimental hadronic analyses.