We conceive a novel channel estimation and data detection scheme for OTFS-modulated faster-than-Nyquist (FTN) transmission over doubly selective fading channels, aiming for enhancing the spectral efficiency and Doppler resilience. The delay-Doppler (DD) domain's input-output relationship of OTFS-FTN signaling is derived by employing a root-raised cosine (RRC) shaping filter. More specifically, we design our DD-domain channel estimator for FTN-based pilot transmission, where the pilot symbol interval is lower than that defined by the classic Nyquist criterion. Moreover, we propose a reduced-complexity linear minimum mean square error equalizer, supporting noise whitening, where the FTN-induced inter-symbol interference (ISI) matrix is approximated by a sparse one. Our performance results demonstrate that the proposed OTFS-FTN scheme is capable of enhancing the achievable information rate, while attaining a comparable BER performance to both that of its Nyquist-based OTFS counterpart and to other FTN transmission schemes, which employ the same RRC shaping filter.