This work analyzes pressure and hydrogen enrichment effects on the stabilization and combustion dynamics of a partially-premixed swirled-stabilized methane flame operated at 1, 3 and 5 bar, with H2 admixture up to 40% by volume. Large Eddy Simulations (LES) are performed to analyze flame stabilization and dynamics for all cases. When pressure is increased, both turbulence and chemical time scales are reduced leading to smaller turbulent scales and thinner flames which makes LES modeling challenging. To ensure that all flames are resolved similarly in the tested pressure range, the dynamic formulation of the thickened flame model (DTFLES) is used here with a Static Mesh Refinement (SMR) strategy. An Analytically Reduced Chemistry (ARC) scheme is employed to describe CH4-H2/Air chemistry. LES is validated against experimental multi-kHz repetition-rate OH* chemiluminescence, OH Planar Laser Induced Fluorescence (PLIF), stereoscopic Particle Image Velocimetry (sPIV) and pressure recordings. The dynamics of the different flames are then addressed. First, the impact of hydrogen at atmospheric pressure is investigated. While the reference natural gas flame (1 bar, 0% H2) presents a lifted M-shape with a strong Precessing Vortex Core (PVC), 40% H2-enrichment modifies the flame which becomes an attached V-shape, with a weakened PVC and the triggering of a thermoacoustic oscillation at the combustion chamber first acoustic mode. Second, the impact of mean pressure is analyzed by fixing the H2-enrichment while increasing the mean pressure to 3 and then 5bar. As the pressure increases, the flame assumes a more compact M-shape. It is proven that the interaction with the turbulent smallest scales is not affected by pressure: Karlovitz number is constant. On the contrary, the flame response to the large turbulent structures is modified: Damkhler number reduces when mean pressure increases. If strong flame/PVC interactions are observed at atmospheric pressure leading to flame tip/root roll-up and local quenching, a more coherent flame is observed when pressure is raised. Furthermore, for these points, the thermoacoustic oscillation coupled with the first chamber acoustic mode rapidly disappears and stable conditions are recovered.