We propose adjustable phase shift pilots (APSPs) for channel acquisition in wideband massive multiple-input multiple-output (MIMO) systems employing orthogonal frequency division multiplexing (OFDM) to reduce the pilot overhead. Based on a physically motivated channel model, we first establish a relationship between channel space-frequency correlations and the channel power angle-delay spectrum in the massive antenna array regime, which reveals the channel sparsity in massive MIMO-OFDM. With this channel model, we then investigate channel acquisition, including channel estimation and channel prediction, for massive MIMO-OFDM with APSPs. We show that channel acquisition performance in terms of sum mean square error can be minimized if the user terminals' channel power distributions in the angle-delay domain can be made non-overlapping with proper phase shift scheduling. A simplified pilot phase shift scheduling algorithm is developed based on this optimal channel acquisition condition. The performance of APSPs is investigated for both one symbol and multiple symbol data models. Simulations demonstrate that the proposed APSP approach can provide substantial performance gains in terms of achievable spectral efficiency over the conventional phase shift orthogonal pilot approach in typical mobility scenarios.