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B[\,e\,] supergiants are a rare and unusual class of massive and luminous stars, characterised by opaque circumstellar envelopes. GG Carinae is a binary whose primary component is a B[\,e\,] supergiant and whose variability has remained unsatisfactorily explained. Using photometric data from ASAS, OMC, and ASAS-SN, and spectroscopic data from the Global Jet Watch and FEROS to study visible emission lines, we focus on the variability of the system at its $\sim$31-day orbital period and constrain the stellar parameters of the primary. There is one photometric minimum per orbital period and, in the emission line spectroscopy, we find a correlation between the amplitude of radial velocity variations and the initial energy of the line species. The spectral behaviour is consistent with the emission lines forming in the primary's wind, with the variable amplitudes between line species being caused by the less energetic lines forming at larger radii on average. By modelling the atmosphere of the primary, we are able to model the radial velocity variations of the wind lines in order to constrain the orbit of the binary. We find that the binary is even more eccentric than previously believed ($e=0.5\pm0.03$). Using this orbital solution, the system is brightest at periastron and dimmest at apastron, and the shape of the photometric variations at the orbital period can be well described by the variable accretion by the secondary of the primary's wind. We suggest that the evolutionary history of GG Carinae may need to be reevaluated in a binary context.