We investigate full-duplex (FD) multi-user multiple input single-output systems with coarse quantization, aiming to characterize the impact of employing low-resolution analog-to-digital converters (ADCs) on self-interference (SI) and to develop a quantization- and SI-aware beamforming method that alleviates quantization-induced performance degradation in the FD systems. We first present an analysis on the perantenna signal-to-quantization noise ratio for conventional linear beamformers to provide the desired range of the number of analog-to-digital converter (ADC) bits, providing system insights for reliable FD operation in regard to the ADC resolution and beamforming strategy. Motivated by the insights, we then propose an SI-aware beamforming method that mitigates residual SI and quantization distortion. The resulting spectral efficiency (SE) maximization problem is decomposed into two tractable subproblems solved via alternating optimization: precoder and combiner design. The precoder optimization is formulated as a generalized eigenvalue problem, where the dominant eigenvector yields the best stationary solution through power iteration, while the combiner is derived as a quantization-aware minimum meansquared error (MMSE) filter. Numerical studies show that the number of required ADC bits with the proposed beamforming falls within the derived theoretical range while achieving the highest SE compared to benchmarks.