  We determine the impact of the $\theta$-angle and axion physics on the near
conformal dynamics of the large-charge baryon sector of $SU(2)$ gauge theories
with $N_f$ fermions in the fundamental representation. We employ an effective
approach featuring Goldstone and dilaton degrees of freedom augmented by the
topological terms in the theory. We investigate how different dilaton
potentials, including the ones for which a systematic counting scheme can be
established, affect the results. Via state-operator correspondence we compute
the corrections to the would-be conformal dimensions of the lowest large-charge
operators as a function of the $\theta$ term and dilaton potential.
