While the geodesic motion around a Kerr black hole is fully integrable and therefore regular, numerical explorations of the phase-space around other spacetime configurations, with a different multipolar structure, have displayed chaotic features. In most of these solutions, the role of the purely general relativistic effect of frame-dragging in chaos has eluded a definite answer. In this work, we show that, when considering neutral test particles around a family of stationary axially-symmetric analytical exact solution to the Einstein-Maxwell field equations, frame-dragging (as captured through spacetime vorticity) is capable of reconstructing KAM-tori from initially highly chaotic configurations. We study this reconstruction by isolating the contribution of the spacetime vorticity scalar to the dynamics, and exemplify our findings by computing rotation curves and the dimensionality of phase-space. Since signatures of chaotic dynamics in gravitational waves have been suggested as a way to test general relativity in the strong-field regime, we have computed gravitational waveforms using the semi-relativistic approximation and studied the frequency content of the gravitational wave spectrum. If this mechanism is generic, chaos suppression by frame-dragging may undermine present proposals to verify the hypothesis of the no-hair theorems and the validity of general relativity.