van der Waals (vdW) interactions play a crucial role in ultracold atom-molecule collisions, as they dominate the long-range part of the interparticle potential and govern essential scattering properties at millikelvin and submillikelvin. In this work, we systematically investigate the vdW coefficient C₆ between alkali-metal atoms and alkali-metal polar diatomic molecules in the ¹Σ⁺ electronic state. Based on secondorder perturbation theory, we derive analytical expressions for the C₆ coefficient for the following cases: an atom in the s, p or d state interacting with a molecule in the rotational ground state (j = 0), and an atom in the s state interacting with a molecule in a rotationally excited state (j > 0). The formulas explicitly include the dependence on the vibrational quantum number ν and rotational quantum number j, enabling a quantitative description of the internal-state dependence of the long-range interaction. Taking the K-NaK system as an example, we calculate the C₆ coefficients using the derived formulas and analyze their variation with vibrational and rotational quantum numbers. Our results reveal that the vibrational excitation significantly enhances the C₆ coefficient, while the rotational excitation shows a relatively weaker but non-negligible effect. Furthermore, to determine the applicable range of the vdW potential, we compare the potential energy curves obtained from perturbation theory with those from a full diagonalization of the dipole-dipole interaction matrix. This comparison allows us to establish the minimum internuclear separation above which the second-order perturbative treatment remains accurate. Our comparison reveals that the vdW potential provides a quantitative description of the long-range interaction for R > 35 a₀ in the K(4s)-NaK(X¹Σ⁺, ν = 0) case. However, for the K(4p)-NaK(X¹Σ⁺, ν = 0) system, the perturbative treatment requires a larger separation (R > 80 a₀) to achieve the same level of accuracy, indicating a significantly extended range of the non-perturbative dipole-dipole coupling. Our findings provide systematic theoretical tools for modeling long-range interactions in alkali atom-molecule systems and provide quantitative reference data for future experiments on ultracold atom-molecule mixtures and quantum gas experiments.