$\newcommand{\de}{\delta}
\newcommand{\ga}{\gamma}
\newcommand{\si}{\sigma}
\newcommand{\ep}{\epsilon}$
Take any real $n>0$ and any $z:=\zeta\in(0,\infty)$. Let $\ep\downarrow0$.
For real $t\ne0$, let $u:=\arctan(\ep/t)$, so that $t=\ep\cot u$.
Then for $t>0$
\begin{equation*}
\left|(t-i\ep)^{-n}-(t+i\ep)^{-n}\right|=2(t^2+\ep^2)^{-n/2}|\sin(nu)|
=2\ep^{-n}|\sin(nu)|\,|\sin u|^n
\end{equation*}
and for $t<0$
\begin{equation*}
\left|(t-i\ep)^{-n}-(t+i\ep)^{-n}\right|
=2\ep^{-n}|\sin(n(\pi+u))|\,|\sin u|^n.
\end{equation*}
So,
\begin{equation*}
I_n(z,\ep)=2\ep^{1-n}(J_n^+(\de)+J_n^-(\de)), \tag{1}
\end{equation*}
where
\begin{equation*}
\de:=\arctan\frac\ep z\downarrow0, \tag{1a}
\end{equation*}
\begin{equation*}
J_n^+(\de):=\int_\de^{\pi/2}|\sin(nu)|(\sin u)^{n-2}\,du
\to J_n^+(0)\in(0,\infty), \tag{2}
\end{equation*}
and
\begin{equation*}
J_n^-(\de):=\int_{-\pi/2}^{-\de}|\sin(n(\pi+u))|\,|\sin u|^{n-2}\,du
=\int_\de^{\pi/2}|\sin(n(\pi-v))|\,(\sin v)^{n-2}\,dv.
\end{equation*}
If $n\ge1$, then $J_n^-(\de)\to J_n^-(0)$ and hence
\begin{equation*}
I_n(z,\ep)\sim2\ep^{1-n}(J_n^+(0)+J_n^-(0))
=2\ep^{1-n}\int_0^{\pi/2}(|\sin(nu)|+|\sin(n(\pi-u))|)(\sin u)^{n-2}\,du.
\end{equation*}
So,
$\displaystyle{I_n(z,\ep)\sim2\ep^{1-n}\int_0^\pi |\sin(nu)|(\sin u)^{n-2}\,du}$ for $n\ge1$.
In particular, for $n=1$ we get $I_n(z,\ep)\to2\pi$.
It remains to consider the case $0<n<1$. Let $a>0$ vary with $\ep$ in any way such that $a\downarrow0$ and $\ep=o(a)$, so that $\de=o(a)$.
then
\begin{equation*}
J_n^-(\de)=\int_\de^a+\int_a^{\pi/2}, \tag{3}
\end{equation*}
where
\begin{equation*}
\int_\de^a:=\int_\de^a |\sin(n(\pi-v))|\,(\sin v)^{n-2}\,dv
\sim\sin(n\pi)\int_\de^a v^{n-2}\,dv
\sim\sin(n\pi)\frac{\de^{n-1}}{1-n}, \tag{4}
\end{equation*}
\begin{equation*}
\Big|\int_a^{\pi/2}\Big|\le\int_a^{\pi/2}v^{n-2}\,dv=o(\de^{n-1}). \tag{5}
\end{equation*}
Collecting the pieces (1), (1a), (2), (3), (4), and (5), we see that
$\displaystyle{I_n(z,\ep)\to2z^{1-n}\frac{\sin(n\pi)}{1-n}}$ for $n\in(0,1)$.
