Is the specific heat of salt the same as Na + CL when dissolved?
Salt does not dissolve to to $\ce{Na}$ and $\ce{Cl}$ atoms, but to hydrated $\ce{Na+}$ and $\ce{Cl-}$ ions, so it's heat capacity is very different to the solid $\ce{NaCl}$.

And yes it looses vibration modes of the crystal lattice, less the rotation, as there is nothing to rotate in the lattice.

There are also vibrations of coordination bonds to water. Dissolved ions also have translational degrees of freedom, what is missing in solids.

Even for the identical substance, its heat capacity in solid and liquid state differ. E.g. liquid water has about 2x bigger heat capacity than ice ( 4.2 versus 2.1 J/g/K ).