  We find Fano threefolds $X$ admitting K\"ahler-Ricci solitons (KRS) with
non-trivial moduli, which are $\mathbb{T}$-varieties of complexity two. More
precisely, we show that the weighted K-stability of $(X,\xi_0)$ (where $\xi_0$
is the soliton candidate) is equivalent to certain GIT-stability. In
particular, this provides the first examples of strictly weighted K-semistable
Fano varieties. On the other hand, we generalize Koiso's theorem to the log
Fano setting. Indeed, we show that the K-stability of a log Fano pair
$(V,\Delta_V)$ is equivalent to the weighted K-stability of a cone $(Y,
\Delta_Y, \xi_0)$ over it. This also leads to new examples of KRS Fano
varieties with non-trivial moduli and small automorphism groups. To achieve
these, we establish the weighted Abban-Zhuang estimate generalizing the work of
\cite{AZ22}, which gives a lower bound of the weighted stability threshold
$\delta^g_{\mathbb{T}}(X,\Delta)$. This is an effective way to check the
weighted K-semistablity of a log Fano triple $(X,\Delta,\xi_0)$. Surprisingly,
such an estimate is also useful in testing (weighted) K-polystability based on
the work of \cite{BLXZ23}.
