The effect of plasma screening on the electronic and vibrational structure of one-electron diatomic molecules is analyzed by including screened Coulomb interactions among all their particles in the solution of the Schrödinger equation. The breakdown of the well-known separability of the Schrödinger equation in confocal elliptical coordinates, the lost of long-range Coulomb interaction and the non-degeneracy in both the united atom and separated atom limits bring new elements into the analysis of the modified binding energies (electronic and vibrational), adiabatic correlation rules, molecular Stark mixings due to polarization effects, unexpected presence of molecular shape resonances and changes in non-adiabatic and radiative couplings upon the variation of the screening strength.