Orthogonal signal-division multiplexing (OSDM) is a recently emerging modulation scheme which, compared to conventional orthogonal frequency-division multiplexing, can effectively lower the peak-to-average power ratio and introduce intra-vector frequency diversity. In this paper, a time-domain oversampled OSDM system for underwater acoustic (UWA) communications is designed, where each OSDM vector is equivalently transmitted over multiple virtual channels, and thus an enhanced frequency diversity gain can be achieved. Moreover, at the receiver, zero vectors and frequency-shifted Chu sequences are used for Doppler compensation and channel estimation, respectively, while low-complexity per-vector equalization is performed based on the composite channel matrix factorization. Finally, the performance of the proposed OSDM system is evaluated through both numerical simulations and a short-range field experiment, and its effectiveness over time-varying UWA channels is confirmed.