Cross-linked networks of semiflexible biopolymers are one of the essential building blocks of life as they are the scaffolds providing mechanical strength to biological cells to handle external stress and regulate shape. These protein structures experience strain at different rates often under confinement such as a membrane. Here, we compute the steady-state dynamics of stress and stress fluctuations in a wall-confined, continuously sheared, reversibly cross-linked, sticker-spacer model of a semiflexible biopolymer network. We find that the averages and fluctuations of shear stress and pressure increase by orders of magnitude when the strain rate is increased above a certain regime. The shear viscosity decreases with increasing strain rate except near the critical strain rate regime where it exhibits an inflection. Upon increasing the strain rate, we note a shift from a long time autocorrelation to an oscillatory and then to a sharply dropping autocorrelation function, endorsed by corresponding changes in the power spectrum of the stress. These outcomes indicate a transition from stick to stick-slip (stress buildup and relaxation) and then to slip upon increasing the strain rate, and we posit that this has to be a hallmark intermittent response of a dynamically cross-linked network under continuous shear deformations. We suggest that a fluctuation-dissipation type framework, where the stress is a stochastic process and "resistance to stress" is a function of strain rate, can help us understand the stress dynamics in biopolymer networks.