Precise ratios of elastic electron scattering cross sections for the nuclei $^{54,56,58}\mathrm{Fe}$, $^{58,60,62,64}\mathrm{Ni}$, and $^{64,66,68,70}\mathrm{Zn}$ have been measured in the momentum-transfer region $0.6 {\mathrm{fm}}^{\ensuremath{-}1}\ensuremath{\le}q\ensuremath{\le}2.3 {\mathrm{fm}}^{\ensuremath{-}1}$. The data were analyzed with a Fourier-Bessel parametrization of the charge distribution. Charge distribution differences were determined nearly model independently for the $\ensuremath{\Delta}A=2$ isotope and isotone pairs. The $\ensuremath{\Delta}Z=2$ isotone charge distribution differences show a strong shell effect, if one compares the Ni-Fe charge distribution differences, where the two added protons close the $1{f}_{\frac{7}{2}}$ shell, and the Zn-Ni charge distribution differences, where the two added protons start to fill the $2{p}_{\frac{3}{2}}$ shell. For the $\ensuremath{\Delta}N=2$ isotopes we observe a nearly constant increase of the "half density radius" in the investigated region of the $2{p}_{\frac{3}{2}}$, $1{f}_{\frac{5}{2}}$, and $2{p}_{\frac{1}{2}}$ neutron shells. However, the skin thickness of the charge distribution increases strongly at the beginning ($^{56}\mathrm{Fe}$-$^{54}\mathrm{Fe}$) and decreases at the end ($^{70}\mathrm{Zn}$-$^{68}\mathrm{Zn}$) of these shells by adding two neutrons. The rms radii differences deduced from a combined analysis of the electron scattering data and present muonic x-ray data show the same trend. These differences, determined model independently with a typical accuracy of some ${10}^{\ensuremath{-}3}$ fm, decrease nearly linearly with increasing neutron number in the investigated region $28\ensuremath{\le}N\ensuremath{\le}40$. These isotope shifts are nearly independent of the proton configuration of the involved nuclei, which indicates that the added neutrons interact primarily with the proton core rather than with the valence protons. Recent calculations, which include ground state correlations, show that the observed isotope shifts reflect deformation changes caused by changes of the amplitude of the zero-point quadrupole surface oscillations. A comparison of the experimental charge distribution differences with results of density dependent Hartree-Fock calculations also indicates the importance of deformation changes. The core rearrangement due to added protons, deduced from the measured isotone charge distribution differences, and the core rearrangements due to added neutrons, directly measured by the isotope shifts, are similar. The last two nucleons of both the $1{f}_{\frac{7}{2}}$ neutron and proton shell cause core polarizations, resulting in a smaller charge core radius.NUCLEAR STRUCTURE $^{54,56,58}\mathrm{Fe}$, $^{58,60,62,64}\mathrm{Ni}$, and $^{64,66,68,70}\mathrm{Zn}$; elastic electron scattering differential cross sections $\frac{d\ensuremath{\sigma}}{d\ensuremath{\Omega}(E, \ensuremath{\theta})}$ and ratios at $E=100, 150, 275$ MeV; model-independent charge distribution differences for isotones and isotopes; model-independent radial moments ${〈{\mathcal{r}}^{k}〉}^{\frac{1}{k}}$ and $\ensuremath{\Delta}{〈{\mathcal{r}}^{k}〉}^{\frac{1}{k}}$, deduced from a combined elastic electron and muonic x-ray data analysis; charge distribution differences, compared with Hartree-Fock calculations, and calculations considering ground state correlations.