Although solid helium-4 (He-4) may be a supersolid, it also exhibits many phenomena unexpected in that context. We studied relaxation dynamics in the resonance frequency f(T) and dissipation D(T) of a torsional oscillator containing solid He-4. With the appearance of the "supersolid" state, the relaxation times within f(T) and D(T) began to increase rapidly together. More importantly, the relaxation processes in both D(T) and a component of f(T) exhibited a complex synchronized ultraslow evolution toward equilibrium. Analysis using a generalized rotational susceptibility revealed that, while exhibiting these apparently glassy dynamics, the phenomena were quantitatively inconsistent with a simple excitation freeze-out transition because the variation in f was far too large. One possibility is that amorphous solid He-4 represents a new form of supersolid in which dynamical excitations within the solid control the superfluid phase stiffness.