A crucial issue in cuprates is the extent and mechanism of the coupling of the lattice to the electrons and the superconductivity. Here we report Cu K edge extended X-ray absorption fine structure measurements elucidating the internal quantum tunneling polaron (iqtp) component of the dynamical structure in two heavily overdoped superconducting cuprate compounds, tetragonal YSr2Cu2.75Mo0.25O7.54 with superconducting critical temperature, Tc = 84 K and hole density p = 0.3 to 0.5 per planar Cu, and the tetragonal phase of Sr2CuO3.3 with Tc = 95 K and p = 0.6. In YSr2Cu2.75Mo0.25O7.54 changes in the Cu-apical O two-site distribution reflect a sequential renormalization of the double-well potential of this site beginning at Tc, with the energy difference between the two minima increasing by ∼6 meV between Tc and 52 K. Sr2CuO3.3 undergoes a radically larger transformation at Tc, >1-Å displacements of the apical O atoms. The principal feature of the dynamical structure underlying these transformations is the strongly anharmonic oscillation of the apical O atoms in a double-well potential that results in the observation of two distinct O sites whose Cu–O distances indicate different bonding modes and valence-charge distributions. The coupling of the superconductivity to the iqtp that originates in this nonadiabatic coupling between the electrons and lattice demonstrates an important role for the dynamical structure whereby pairing occurs even in a system where displacements of the atoms that are part of the transition are sufficiently large to alter the Fermi surface. The synchronization and dynamic coherence of the iqtps resulting from the strong interactions within a crystal would be expected to influence this process.