Orthogonal time frequency space (OTFS) waveform provides resilience against doubly dispersive wireless channel resulting in high mobility application. This paper amalgamates OTFS waveform with full-duplex (FD) distributed massive multi-ple input multiple output (mMIMO) communications to achieve high throughput gain in high mobility applications. In particular, we consider an OTFS-aided FD cell-free mMIMO system, where FD access points (APs) simultaneously serve half-duplex (HD) uplink (UL) and downlink (DL) users. We derive novel closed-form spectral efficiency (SE) expressions for both UL and DL in the existence of channel estimation errors. Our simulations validate the correctness of the derived closed-form SE expressions and highlight the impact of velocity on sum SE for two different vehicular models. The simulation results also demonstrate the SE improvement of the FD cell-free mMIMO-OTFS system over its HD counterpart.