We present results for the isovector and flavor diagonal tensor charges , , , and needed to probe novel tensor interactions at the TeV scale in neutron and nuclear -decays and the contribution of the quark electric dipole moment (EDM) to the neutron EDM. The lattice QCD calculations were done using nine ensembles of gauge configurations generated by the MILC collaboration using the HISQ action with dynamical flavors. These ensembles span three lattice spacings , 0.09 and 0.12 fm and three quark masses corresponding to the pion masses , 220 and 310 MeV. Using estimates from these ensembles, we quantify all systematic uncertainties and perform a simultaneous extrapolation in the lattice spacing, volume and light quark masses for the connected contributions. The final estimates of the connected nucleon (proton) tensor charge for the isovector combination is in the scheme at 2 GeV. The additional disconnected quark loop contributions needed for the flavor-diagonal matrix elements are calculated using a stochastic estimator employing the truncated solver method with the all-mode-averaging technique. We find that the size of the disconnected contribution is smaller than the statistical error in the connected contribution. This allows us to bound the disconnected contribution and include it as an additional uncertainty in the flavor-diagonal charges. After a continuum extrapolation, we find , and . The strangeness tensor charge, that can make a significant contribution to the neutron EDM due to the large ratio , is in the continuum limit.