Oscillating airfoils serve as a platform to study the flapping and undulating motions employed by flying and swimming organisms to generate lift and thrust. These motions intentionally produce separated flows that shed vortices as a consequence of the underlying force production. Classical unsteady aerodynamic studies often focus on identifying efficient performance regimes and then analyzing the accompanying vortex dynamics. While this work includes elements of this approach, the principal objective is to use these flow experiments as a backdrop for investigating the dynamics and impulse of vorticity. The experiments conducted here occur in the Cyber-Physical Fluid Dynamics (CPFD) facility, a hybrid water-channel and towing-tank. In the first part of this work, the airfoil self-propels freely with a rotational pitch bias that breaks the mirrored symmetry of the motion. We perform a parameter sweep on kinematics and identify the characteristic vortex shedding patterns that govern a Pareto-efficient regime. The second and third parts of this work depart from a canonical unsteady aerodynamic study and investigate vortex impulse in these oscillating airfoil flows. We decompose periodic spatiotemporal vorticity fields into oscillatory modes using Dynamic Mode Decomposition (DMD) and show that the vortex impulse equation can be interpreted on a mode-by-mode basis. By varying flow kinematics to induce changes in vortex shedding behavior, we illustrate the relationship between modal force and vortex dynamics. Finally, we employ a reduced-order discrete vortex impulse model to estimate the transverse force amplitude of a symmetrically oscillating airfoil. The simplicity of the model allows the amplitude to be estimated from only the bulk properties (circulation and velocity) of a free vortex in the wake. We apply physical intuition of the production, transport, and decay of vorticity to interpret the extent of the model’s validity.