The chiral magnetic wave (CMW) has been theorized to propagate in the deconfined nuclear medium formed in high-energy heavy-ion collisions and to cause a difference in elliptic flow (v2) between negatively and positively charged hadrons. Experimental data consistent with the CMW have been reported by the STAR Collaboration at the Relativistic Heavy Ion Collider (RHIC), based on the charge asymmetry dependence of the pion v2 from Au+Au collisions at sNN=27 to 200 GeV. In this comprehensive study, we present the STAR measurements of elliptic flow and triangular flow of charged pions, along with the v2 of charged kaons and protons, as a function of charge asymmetry in Au+Au collisions at sNN=27, 39, 62.4, and 200 GeV. The slope parameters extracted from the linear dependence of the v2 difference on charge asymmetry for different particle species are reported and compared in different centrality intervals. In addition, the slopes of v2 for charged pions in small systems, i.e., p+Au and d+Au at sNN=200 GeV, are also presented and compared with those in large systems, i.e., Au+Au at sNN=200 GeV and U+U at 193 GeV. Our results provide new insights for the possible existence of the CMW and further constrain the background contributions in heavy-ion collisions at RHIC energies.