Flow dynamics of gaseous-oxygen/kerosene jet-swirl injectors are systematically investigated at supercritical pressure. A unified theoretical and numerical framework is implemented in accordance with real-fluid thermodynamics. Turbulent closure is achieved with large eddy simulation. The appropriate numerical resolution is ensured through a grid independence study. GOX at 687K is injected into the inner jet, while kerosene at 492.2K is tangentially introduced into the outer swirler. Dynamic responses of the flowfield are explored using the spectral analysis and proper orthogonal decomposition. Longitudinal acoustics are found to be dominant in the GOX post and recess region, while the shear-layer instability dominates in the taper and downstream region. Effect of recess length is studied for the dominant mechanisms of flow instabilities. As the recess length decreases, the frequencies of acoustic instability remain constant, while the frequencies for shear-layer instability increase. For shielded cases, dominant frequencies in the GOX post and the taper region are distinct, due to acoustic damping nature of taper region. For unshielded cases, the hydrodynamic instability is excited by longitudinal modes since kerosene is directly impinging upon the GOX during early stage of flow development, leading to identical frequency in both GOX post and combustion chamber.