We present experimental results for spherical particles rising and settling in a still fluid. Imposing a well-controlled center of mass offset enables us to vary the rotational dynamics selectively by introducing an intrinsic rotational timescale to the problem. Results are highly sensitive even to small degrees of offset, rendering this a practically relevant parameter by itself. We further find that, for a certain ratio of the rotational to a vortex shedding timescale (capturing a Froude-type similarity), a resonance phenomenon sets in. Even though this is a rotational effect in origin, it also strongly affects translational oscillation frequency and amplitude, and most importantly, the drag coefficient. This observation equally applies to both heavy and light spheres, albeit with slightly different characteristics for which we offer an explanation. Our findings highlight the need to consider rotational parameters when trying to understand and classify path properties of rising and settling spheres.