This paper presents two novel hybrid beamforming (HYBF) designs for a multi-cell massive multiple-input-multiple-output (mMIMO) millimeter wave (mmWave) full duplex (FD) system under limited dynamic range (LDR). Firstly, we present a novel centralized HYBF (C-HYBF) scheme based on alternating optimization. In general, the complexity of C-HYBF schemes scales quadratically as a function of the number of users and cells, which may limit their scalability. Moreover, they require significant communication overhead to transfer complete channel state information (CSI) to the central node every channel coherence time for optimization. The central node also requires very high computational power to jointly optimize many variables for the uplink (UL) and downlink (DL) users in FD systems. To overcome these drawbacks, we propose a very low-complexity and scalable cooperative per-link parallel and distributed (P$\&$D)-HYBF scheme. It allows each mmWave FD base station (BS) to update the beamformers for its users in a distributed fashion and independently in parallel on different computational processors. The complexity of P$\&$D-HYBF scales only linearly as the network size grows, making it desirable for the next generation of large and dense mmWave FD networks. Simulation results show that both designs significantly outperform the fully digital half duplex (HD) system with only a few radio-frequency (RF) chains and achieve similar performance.