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Using the diabatic formalism, which generalizes the adiabatic approximation in the Born-Oppen-heimer formalism, we apply well-known Hamiltonian methods to calculate the effect of open di-meson thresholds that lie well below the mass of elementary $c\overline{c}q{\overline{q}}^{\prime }$, $c\overline{c}s\overline{s}$, and $c\overline{c}q\overline{s}$tetraquark bound states. We compute the resulting mass shifts for these states, as well as their decay widths to the corresponding meson pairs. Each mass eigenstate, originally produced using a bound-state approximation under the diabatic formalism, consists of an admixture of a compact diquark-antidiquark configuration (an eigenstate of the original dynamical diquark model) with an extended di-meson configuration induced by the nearest threshold. We compare our results with those from our recent work that employs a scattering formalism, and find a great deal of agreement, but also comment upon interesting discrepancies between the two approaches. Published by the American Physical Society2024