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We report on the observation of nonlinear dynamics in driven-dissipative Bose-Hubbard dimers using a pair of coupled fiber ring resonators coherently driven by a single laser beam. We comprehensively explore the optical switching arising when scanning the detuning of the undriven cavity and demonstrate the key role played by the driven cavity detuning in the nonlinear regime. By driving the photonic dimer out of equilibrium, we observe the occurrence of robust self-switching oscillations.