We investigate the dynamics of single- and multi-photon emission from detuned strongly coupled systems based on the quantum-dot-photonic-crystal resonator platform. Transmitting light through such systems can generate a range of non-classical states of light with tunable photon counting statistics due to the nonlinear ladder of hybridized light-matter states. By controlling the detuning between emitter and resonator, the transmission can be tuned to strongly enhance either single- or two-photon emission processes. Despite the strongly-dissipative nature of these systems, we find that by utilizing a self-homodyne interference technique combined with frequency-filtering we are able to find a strong two-photon component of the emission in the multi-photon regime. In order to explain our correlation measurements, we propose rate equation models that capture the dominant processes of emission both in the single- and multi-photon regimes. These models are then supported by quantum-optical simulations that fully capture the frequency filtering of emission from our solid-state system.