A theoretical construction of an antiferromagnetic polymer multilayered field-effect transistor with polymers stretched between the source and drain contacts was undertaken. The model employed a quantum approach to the on-chain spin-charge distribution, which was self-consistently coupled with the charge distribution controlled by the gate voltage. Contrary to standard field-effect transistors, we found that the current firstly increased superlinearly with the drain voltage, then it achieved the maximum for drain voltages notably lower than the gate voltage, and after that, it decreased with the drain voltage with no saturation. Such effects were coupled with the formation of the current spin-polarization ratio, where the on-chain mobility of respective spin-polarized charges was significantly dependent on the applied drain voltage. These effects arise from competition among the antiferromagnetic coupling, the intra-site spin-dependent Coulomb interaction, and the applied drain and gate voltages, which strongly influence the on-chain spin-charge distribution, varying from an alternating spin configuration to a spin-polarized configuration at both ends of the chain. Substantial control over the magnitude of spin-polarized currents was achieved by manipulating gate and drain voltages, showcasing the feasibility of practical applications in spintronics.