Many strongly polar molecules can form an anion by attaching an electron to either an empty or half-filled valence-bound (VB) orbital or a so-called dipole-bound (DB) orbital. These two families of orbitals can be very different in their radial extent (the former are usually more compact, while the latter are quite diffuse) and in the degree to which they are affected by surrounding solvent molecules. In this study, the effects of hydration (representative of strong solvation) on the DB state of a model polar species are investigated with an eye toward determining whether this state is stabilized or even persists when a few to 100 water molecules surround the polar molecule. It is found that in the presence of up to ca. 10-12 water molecules, the excess electron can remain in a DB orbital. However, once there are enough water molecules to form a complete first hydration shell (or more), the excess electron migrates into an orbital localized on the outer surface of the water solvent cage. These findings have implications on the possible role of DB states as doorways to facilitating electron attachment and subsequent electron transfer to VB states. It is shown that even when the electron is bound to the surface of the surrounding solvent, the dipole potential of the solute molecule can influence where on the surface the electron binds. It is also illustrated that using continuum dielectric methods to describe the hydration of DB states is fraught with danger because much of the outermost electron density in such states penetrates outside the boundary of the cavity used in these methods.