In this talk, I review briefly a scenario for the evolution of a string-inspired cosmological model, in which condensates of primordial gravitational waves (GW), formed at the very early eras after the Big Bang, are considered responsible for inducing inflation and then a smooth exit to a radiation dominated epoch. Primordial axion fields, that exist in the fundamental massless gravitational (bosonic) string multiplet, couple to the non-trivial GW-induced anomalies. As a result of this coupling, there exist axion background configurations which violate (spontaneously) Lorentz symmetry, and remain undiluted at the end of inflation. In models with heavy sterile right-handed neutrinos (RHN), such backgrounds are linked to novel (Lorentz and CPT Violating) mechanisms for the generation of matter-antimatter asymmetry in the Cosmos, via the asymmetric decays of the RHN to standard model particles and antiparticles. During the QCD epoch, the axions develop an instanton-induced mass and can, thus, play the r\^ole of Dark Matter (DM). The energy density of such a Universe, throughout its evolution, has the form of that of a ``running vacuum model'', that is, it can be expanded in power series of even powers of the Hubble parameter $H(t)$. The coefficients of those terms, though, are different for the various cosmological epochs. For the phenomenology of our model, which is consistent with the current cosmological data, and could also help in alleviating (some of) the tensions, it suffices to consider up to and including quartic powers of $H(t)$. In the early Universe phase, it is the $H^4(t)$ term, induced by the GW condensate of the gravitational anomaly, that drives inflation without the need for external inflaton fields.