Full Duplex (FD) radio has emerged as a promising solution to increase the data rates by up to a factor of two via simultaneous transmission and reception in the same frequency band. This paper studies a novel hybrid beamforming (HYBF) design to maximize the weighted sum-rate (WSR) in a single-cell millimeter wave (mmWave) massive multiple-input-multiple-output (mMIMO) FD system. Motivated by practical considerations, we assume that the multi-antenna users and hybrid FD base station (BS) suffer from the limited dynamic range (LDR) noise due to non-ideal hardware and an impairment aware HYBF approach is adopted by integrating the traditional LDR noise model in the mmWave band. In contrast to the conventional HYBF schemes, our design also considers the joint sum-power and the practical per-antenna power constraints. A novel interference, self-interference (SI) and LDR noise aware optimal power allocation scheme for the uplink (UL) users and FD BS is also presented to satisfy the joint constraints. The maximum achievable gain of a multi-user mmWave FD system over a fully digital half duplex (HD) system with different LDR noise levels and numbers of the radio-frequency (RF) chains is investigated. Simulation results show that our design outperforms the HD system with only a few RF chains at any LDR noise level. The advantage of having amplitude control at the analog stage is also examined, and additional gain for the mmWave FD system becomes evident when the number of RF chains at the hybrid FD BS is small.