Two-dimensional sigma models on superspheres $S^{r−1|2s}$ $\cong OSp(r|2s)/$ $OSp(r − 1|2s)$ are known to flow to weak coupling $g_\sigma$$\rightarrow$ 0 in the IR when $r  −  2s  <  2$. Their long-distance properties are described by a free ‘Goldstone’ conformal field theory with $r  −  1$ bosonic and 2s fermionic degrees of freedom, where the $OSp(r 2s)$ symmetry is spontaneously broken. This behavior is made possible by the lack of unitarity. The purpose of this paper is to study logarithmic corrections to the free theory at small but non-zero coupling $g_\sigma$. We do this in two ways. On the one hand, we perform perturbative calculations with the sigma model action, which are of special technical interest since the perturbed theory is logarithmic. On the other hand, we study an integrable lattice discretization of the sigma models provided by vertex models and spin chains with $OSp(r 2s)$ symmetry. Detailed analysis of the Bethe equations then confirms and completes the field theoretic calculations. Finally, we apply our results to physical properties of dense loop soups with crossings.