The electrons of an isolated sodium chloride bound pair in vacuum reside at a semi-classical level in their ground state, so that the Born-Oppenheimer approximation applies, and the 'molecule' acts like a sort of rigid rotor, whose relevant degrees of freedom are translational momentum and orbital angular momentum. In solution, however, one might expect things to become a bit more complicated. Electrons and holes from water, H+ and OH- and other metastable arrangements would tend to perturb the electronic structure of the NaCl molecule, and it seems that this could greatly impact its effective behavior (after integrating out the water molecules, say.) It does not seem totally unreasonable that the NaCl pair, once in solution, would spend most of its time partially dissociated, and that the constitutive ions would tend to be far enough from each other to be described reasonably well by a 'semi-classical' Langevin dynamical model. However, if one simulates overdamped Langevin dynamics of two charged ions separated by distances that are within the screening length, one finds that the ions tend to approach each other arbitrarily closely. Because of this, it would seem that the ions must also spend an appreciable time in some sort of quantum mechanical bound state. My question is, to what extent is the structure of this bound state known (beyond that it probably resembles the bound state in vacuum), and if so, what are the relevant 'soft' degrees of freedom?
From the comments, it would seem that NaCl, while existant, is a not-especially-stable molecule that in solution tends to dissociate rapidly, with bare charges screened either with readily available (if intermittent and short-lived) water-based ions or even more readily available aggregate dipole distributions. This would allow the ions to be treated effectively as simple uncoupled Brownian walkers at large scales. I am still not entirely convinced, however, that this picture is adequate when the ionic concentration is low enough that the water is only weakly conductive. Presumably, the 'hydrolysis'(?) of NaCl triggers a cascade of similar dissociations until the free charge recombines elsewhere. If the medium isn't fully conductive, then hydrolysis must produce an effective dipole pair or aggregate/collective dipole moment somewhere around the original source (or, in non-idealized conditions, at least drive the recombination of an existing dipole nearby.)
It might be more sensible to recast the question in terms of the 'dissolution' of (dilute) salt in a more stable solvent (e.g. oil or liquid carbon dioxide or what have you.) Presumably there is a somewhat subtle interplay or trade-off between the electronic structure of the solvent near the salt and the Coulombic attraction between oppositely charged ions. If the solvent is sufficiently stable, then the screening length could be quite large, and it isn't obvious to me that the semi-classical coupled diffusive behavior discussed in the original post would be physically irrelevant.
One could investigate this question in detail using density functional theory, but at a heuristic level it might be enough that at very small length scales (and sufficiently low temperatures, near the melting point), the 'core' degrees of freedom of the solvent would, in its quantum bound state, tend to exhibit gel-like behavior, and that the ionic 'molecule'/pair would act as a sort of inclusion or notch (or inclusion/notch 'pair') in the gel, perhaps with a 'trail' that would depend on how the ions approach the bound state, and which might gradually dissipate as the solvent molecules adjust to the neutral inclusion (and at a faster rate at higher temperatures.) Dissociation of the ions might then be driven by 'micro-quakes' in the solvent matrix (and/or cosmic rays), as ambient oscillations concentrate stress at the inclusion. In any event, vibrational modes are probably more relevant in solution than in vacuum, and the 'bound state' is probably kinetically frustrated or arrested by solid-like behavior of the 'solvent matrix' at small length scales.