  Experiments with paramagnetic ground or metastable excited states of
molecules (ThO, HfF$^+$, YbF, YbOH, BaF, PbO, etc.) provide strong constraints
on electron electric dipole moment (EDM) and coupling constant $C_{SP}$ of
contact semileptonic interaction. We compute new contributions to $C_{SP}$
arising from the nucleon EDMs due to combined electric and magnetic
electron-nucleon interaction. This allows us to improve limits from the
experiments with paramagnetic molecules on the $CP$-violating parameters, such
as the proton EDM, $|d_p|< 1.1\times 10^{-23} e\cdot $cm, the QCD vacuum angle,
$|\bar \theta|<1.4\times 10^{-8}$, as well as the quark chromo-EDMs and
$\pi$-meson-nucleon couplings. Our results may also be used to search for the
axion dark matter which produces oscillating $\bar\theta$.
