Wave-like patterns play fundamental roles in living systems, from single cells to cell collectives, organizing essential functions from wound repair to morphogenesis. While biochemical waves are well established, it remains unclear how the force propagation of cytoskeletal contraction waves regulates cell morphology and migration. Here, we investigate a generic mechanism by which an active droplet exhibits diverse morphodynamics with contraction waves, using a minimal numerical model relevant to living and synthetic cells. We develop a theoretical model in which active gels, representing the active cytoskeleton, generate contraction waves inside a deformable phase-field droplet. The feedback between contraction and deformation organizes cell-like motility modes and revolving contraction waves with unique symmetries. We demonstrate that time-dependent confinement acts as an external driving frequency, entraining contraction waves into a mode-locked revolving state. Our results provide a dynamical-systems perspective on how the coupling of contraction waves and shape deformation of active gel droplets coordinates diverse morphodynamics, a principle relevant to non-equilibrium systems under confinement from living cells to lifelike synthetic cells.