By means of quantum tensor network calculations, we investigate the large Purcell effect experienced by an organic molecule placed in the vicinity of a plasmonic nanostructure. In particular, we consider a donor-π bridge-acceptor dye at the gap of two Ag nanospheres. Our theoretical approach allows for a realistic description of the continua of both molecular vibrations and optical nanocavity modes. We analyze both the exciton dynamics and the corresponding emission spectrum, showing that these magnitudes are not accurately represented by the simplified models used up to date. By disentangling the molecule coupling to radiative and non-radiative plasmonic modes, we also shed light into the quenching phenomenology taking place in the system.