In this work, we systematically investigate the strong coupling constants for vertices involving doubly charmed baryons ( \calB_\rmcc), singly charmed sextet baryons ( \calB_\rmc^6), and D mesons, using the light-cone sum rules 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 \calB_\rmc^6. Through graphical analysis, we 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 the decay constants. To extrapolate the b-baryon light-cone distribution amplitudes to c-baryons, we 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 \rmf_\calB_\rmcc and \rmf_\calB_\rmc^6 dominate over those from other parameters, which can be attributed to the significant inherent uncertainties of these input parameters. The light-cone sum rules results deviate from the ratios predicted by SU(3) flavor symmetry, which are primarily due to the significant differences in the light-cone wave functions of baryons with different light quarks (u, d, s) in the light-cone sum rules 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 quantum chromodynamics sum rules or lattice quantum chromodynamics. 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.