One of the key mechanisms employed by cells to control their spatiotemporal organization is the formation and dissolution of phase-separated condensates. Such balance between condensate assembly and disassembly can be critically regulated by the presence of RNA. In this work, we use a novel chemically accurate coarse-grained model for proteins and RNA to unravel the impact of poly-uridine RNA in modulating the protein mobility and stability within different biomolecular condensates. We explore the behavior of FUS, hnRNPA1 and TDP-43 proteins along with that of their corresponding prion-like domains and RNA-recognition motifs, from absence to moderately high RNA concentration. By characterising the phase diagram, key molecular interactions, surface tension and viscoelastic properties, we report a dual RNA-induced behavior: On the one hand, poly-uridine enhances phase separation at low concentration, whilst at high concentration, it inhibits the ability of proteins to self-assemble. On the other hand, as a consequence of such stability modulation, the viscoelastic liquid properties of the condensates are significantly enhanced at moderately high RNA concentration, as long as the length of poly-uridine strands is comparable or moderately shorter than those of the proteins, whereas protein self-diffusion barely depends on poly-uridine length. On the whole, our work elucidates the different routes by which RNA can regulate phase separation and condensate dynamics, as well as the subsequent aberrant rigidification implicated in the emergence of various neuropathologies and age-related diseases.