Given a square box $Λ_n\subseteq\mathbb Z^2$ of side length $L^n$ with $L,n>1$, we study hierarchical random fields $\{φ_x\colon x\inΛ_n\}$ with law proportional to ${\rm e}^{\frac12β(φ,Δ_nφ)}\prod_{x\inΛ_n}ν({\rm d}φ_x)$, where $β>0$ is the inverse temperature, $Δ_n$ is a hierarchical Laplacian on $Λ_n$, and $ν$ is a non-degenerate $1$-periodic measure on $\mathbb R$. Our setting includes the integer-valued Gaussian field (a.k.a. DG model or Villain Coulomb gas) and the sine-Gordon model. Relying on renormalization group analysis we derive sharp asymptotic formulas, in the limit as $n\to\infty$, for the covariance $\langleφ_xφ_y\rangle$ and the fractional charge $\langle {\rm e}^{2π{\rm i}α(φ_x-φ_y)}\rangle$ in the subcritical $β<β_{\rm c}:=π^2/\log L$, critical $β=β_{\rm c}$ and slightly supercritical $β>β_{\rm c}$ regimes. The field exhibits logarithmic correlations throughout albeit with a distinct $β$-dependence of both the covariance scale and the fractional-charge exponents in the sub/supercritical regimes. Explicit logarithmic corrections appear at the critical point.