This paper introduces a novel Reference Signal and a channel estimation and equalization technique for Zero-Padded waveforms, like ZP-OFDM and the recently proposed FM-OFDM, under doubly-dispersive channels with Doppler and phase noise. We describe a two-stage pilot structure aimed to separately capture the long-term and short-term channel variations and a piecewise estimation and equalization technique, based on approximation of the time-varying channel impulse response by a set of time-invariant channel responses, which are independently equalized and further combined to compensate the channel’s dispersion. Design criteria for the proposed Reference Signal are also given. Numerical results under high phase noise show that piecewise-equalized ZP-OFDM can outperform MMSE-equalized CP-OFDM with CPE compensation, thus avoiding the need of an additional Reference Signal for phase noise mitigation. Results under high mobility demonstrate the superiority of piecewise-equalized FM-OFDM and ZP-OFDM waveforms over MMSE-equalized OTFS and CP-OFDM, outperforming also the highly complex DFE-equalized OTFS for certain modulation orders, without the need to estimate any Doppler components. The degradation incurred by the proposed realistic piecewise estimation technique with respect to ideal estimation depends on the richness of the channel’s multipath profile rather than its Doppler spread.