Various theories beyond the standard model predict new interactions mediated by new light particles with very weak couplings to ordinary matter. Interactions between polarized electrons and unpolarized nucleons proportional to g_{V}^{N}g_{A}^{e}σ[over →]·v[over →] and g_{A}^{N}g_{A}^{e}σ[over →]·v[over →]×r[over →] are two such examples, where σ[over →] is the spin of the electrons, r[over →] and v[over →] are position and relative velocity between the polarized electrons and nucleons, g_{V}^{N}/g_{A}^{N} is the vector or axial-vector coupling constant of the nucleon, and g_{A}^{e} is the axial-vector coupling constant of the electron. Such interactions involving a vector or axial-vector coupling g_{V}^{N}/g_{A}^{N} at one vertex and an axial-vector coupling g_{A}^{e} at the polarized electron vertex can be induced by the exchange of spin-1 bosons. We report new experimental upper limits on such exotic spin-velocity-dependent interactions of the electron with nucleons from dedicated experiments based on a recently proposed scheme. We rotationally modulated two ∼6 Kg source masses at a frequency of 20 Hz. We used four identical atomic magnetometers in an array form to increase the statistics and cancel the common-mode noise. We applied a data processing method based on high precision numerical integration for the four harmonic frequencies of the signal. We reverse the rotation direction of the source masses to flip the signal due to the new interactions; thus, we can apply the [+1,-3,+3,-1] weighting method to remove possible slow drifting. Our constraint on the product of vector and axial-vector couplings is |g_{V}^{N}g_{A}^{e}|<2.1×10^{-34} and on the product of axial-vector and axial-vector couplings is |g_{A}^{N}g_{A}^{e}|<2.4×10^{-22} for an interaction range of 10 m. The new constraints on vector-axial-vector interaction improved by as much as more than 4 orders of magnitude and on axial-axial interaction by as much as 2 orders of magnitude in the corresponding interaction range, respectively.