While the interplay between liquid-liquid phase separation (LLPS) and glass formation in biological systems is highly relevant for their structure formation and thus function, the exact underlying mechanisms are not well known. The kinetic arrest originates from the slowdown at the molecular level, but how this propagates to the dynamics of microscopic phase domains is not clear. Since with diffusion, viscoelasticity, and hydrodynamics, distinctly different mechanisms are at play, the dynamics needs to be monitored on the relevant time and length scales and compared to theories of phase separation. Using x-ray photon correlation spectroscopy, we determine the LLPS dynamics of a model protein solution upon low temperature quenches and find distinctly different dynamical regimes. We observe that the early stage LLPS is driven by the curvature of the free energy and speeds up upon increasing quench depth. In contrast, the late stage dynamics slows down with increasing quench depth, fingerprinting a nearby glass transition. The dynamics observed shows a ballistic type of motion, implying that viscoelasticity plays an important role during LLPS. We explore possible explanations based on the Cahn-Hilliard theory with nontrivial mobility parameters and find that these can only partially explain our findings.