We introduce a guard time-free approach for simultaneous channel estimation and equalization, addressing both inter-carrier interference (ICI) and inter-symbol interference (ISI) in multi-carrier communication systems. The proposed method utilizes two maximum a-posteriori (MAP) equalizers based on the BCJR algorithm, operating in the time-frequency domain. These equalizers share soft-bit data among themselves and with a channel decoder. We derive the a-posteriori probabilities (APPs) and show that the equalizers exploit Dopplertime diversity. We also derive maximum-likelihood (ML) bounds on the system's performance, which also exhibit diversity gains. We initially simulate on Rayleigh fading doubly-dispersive (DD) channels with perfect receiver channel state information (CSI). We compare with an existing iterative maximum likelihood equalizer (IMLE) from the literature and show that the proposed system outperforms it. Next, we focus on an underwater acoustic channel (UAC), and develop a joint channel estimationequalization architecture. To estimate this sparse channel, we take a compressed sensing (CS) approach using the orthogonal matching pursuit (OMP) algorithm together with an overcomplete dictionary set. Compared to 1D equalization with full guard time, the guard time-free estimation-equalization system achieves bit error rate (BER) reductions between 24% and 78% for high-spread channels that can surpass the orthogonal timefrequency space (OTFS) crystallization condition.