Recent advances in the study of nodal Weyl fermions (WFs), quasi-relativistic massless particles, constitute a novel realm of quantum many-body phenomena. The Coulomb interaction in such systems, having a zero density of states at the Fermi level, is of particular interest, since in contrast to conventional correlated metals, its long-ranged component is unscreened. Here, through nuclear-magnetic-resonance (NMR) measurements, we unveil the exotic spin correlations of two-dimensional WFs in an organic material, causing a divergent increase of the Korringa ratio by a factor of 1000 upon cooling, in striking contrast with conventional metallic behaviors. Combined with model calculations, we show that this divergence stems from the interaction-driven velocity renormalization that almost exclusively suppresses the zero-momentum spin fluctuations. At low temperatures, the NMR rate shows a remarkable increase, which is shown by numerical analyses to correspond to inter-node excitonic fluctuations, precursor of a transition from massless to massive quasiparticles.