Heβ spectral line shapes are important for diagnosing temperature and density in many dense plasmas. This work presents Heβ line shapes measured with high spectral resolution from solid-density plasmas with minimized gradients. The line shapes show hallmark features of Stark broadening, including quantifiable redshifts and double-peaked structure with a significant dip between the peaks; these features are compared to models through a Markov chain Monte Carlo framework. Line shape theory using the dipole approximation can fit the width and peak separation of measured line shapes, but it cannot resolve an ambiguity between electron density n_{e} and ion temperature T_{i}, since both parameters influence the strength of quasistatic ion microfields. Here a line shape model employing a full Coulomb interaction for the electron broadening computes self-consistent line widths and redshifts through the monopole term; redshifts have different dependence on plasma parameters and thus resolve the n_{e}-T_{i} ambiguity. The measured line shapes indicate densities that are 80-100% of solid, identifying a regime of highly ionized but well-tamped plasma. This analysis also provides the first strong evidence that dense ions and electrons are not in thermal equilibrium, despite equilibration times much shorter than the duration of x-ray emission; cooler ions may arise from nonclassical thermalization rates or anomalous energy transport. The experimental platform and diagnostic technique constitute a promising new approach for studying ion-electron equilibration in dense plasmas.