Layered A-type antiferromagnetic (AFM) semiconductors offer a unique platform for spintronic devices by combining robust magnetism with gate-tunable semiconducting properties. We present a dual-gate field-effect transistor based on CrSBr, a layered A-type AFM with pronounced transport anisotropy and complex spin dynamics. Our device provides independent control over carrier density and displacement field. Systematic transport measurements reveal that the displacement field, rather than doping, predominantly governs the sign and magnitude of magnetoresistance ratio (MRR). We identify the origins of both negative and positive MRR components and connect them to interlayer and intralayer spin-dependent transport mechanisms. We interpret these findings with ab initio calculations and an effective spin-circuit model that reproduces the anisotropic magnetotransport. Our results identify the displacement field as an efficient electrical control parameter of spin transport in CrSBr and suggest that multi-gate architectures could provide similar control in other layered antiferromagnetic semiconductors.