Woooo, I think I finally find a proof!$
\newcommand{\Lie}{\operatorname{Lie}}
\newcommand{\Lg}{\mathfrak{g}}
\newcommand{\Ll}{\mathfrak{l}}
\newcommand{\Ad}{\operatorname{Ad}}$

Let $G=\operatorname{Aut}(\Lg)^\circ$ be a corresponding complex Lie group. Write $X=X_s+X_n$ the Jordan decomposition into semisimple and nilpotent part. The centralizer $L:=Z_G(X_s)$ is a Levi subgroup. Denote by $\Ll=\Lie L$ and $Z(\Ll)=\Lie Z(L)$ the center of $\Ll$. Write $\mathcal{O}:=Z(\Ll)\cdot\Ad(L)X_n$, a locally closed subvariety in $\Ll$. Note that $\mathcal{O}\cong Z(\Ll)\times\Ad(L)X_n$ has an obvious map to $Z(\Ll)$.

Pick a parabolic subgroup $P$ containing $L$ as its Levi subgroup. Let $U$ be the unipotent radical of $P$ so that $P=LU$. There exists a nilpotent $G$-orbit in $\Lg$ that intersects $X_n+\Lie U$ at a dense open subset of $X_n+\Lie U$. Let $e$ be any element in the intersection. We have to prove the two asserted property in the question, i.e.

- $\Ad(G)X$ meets the Slodowy slice $e+Z_{\Lg}(f)$ and
- $\dim Z_G(X)=\dim Z_G(e)$.

We have $Z_G(X)=Z_L(X_n)$ and also $\dim Z_G(e)=\dim Z_L(X_n)$ [LS79, Theorem 1.3]. This proves (2).

Consider the “generalized Grothendieck-Springer alternation” given by
$$
\tilde{\mathfrak{g}}_X:=\{(g,\gamma)\in (P\backslash G)\times\mathfrak{g}\;|\;\Ad(g)\gamma\in\Lie P\text{ is such that its image under}\Lie P\twoheadrightarrow\Lie L\text{ lies in }\mathcal{O}\}.
$$
It is the usual Grothendieck–Springer alternation if $X$ is regular semisimple, hence the name. There is a natural smooth map $\mu:\tilde{\mathfrak{g}}_X\rightarrow Z(\Ll)$ via $\mathcal{O}\twoheadrightarrow Z(\Ll)$ and another natural $G$-equivariant map $\pi:\tilde{\Lg}_X\rightarrow\Lg$ sending $(g,\gamma)\mapsto\gamma$. For $\zeta\in Z(\Ll)$ that are $(G/L)$-regular (this means $Z_G(\zeta)=L$), we have $\pi(\mu^{-1}(\zeta))=\Ad(G)(\zeta+X_n)$. For the central fiber we have instead $\pi(\mu^{-1}(0))\supset\Ad(G)e$.

Since $e+Z_{\Lg}(f)$ meets $\Ad(G)e$ transversally, there exists a neighborhood (analytically or algebraically, whichever) $U$ of $e$ in $\Lg$ such that for any orbit $\Ad(G)Y$ we have $\Ad(G)Y\cap U\not=\emptyset\implies\Ad(G)Y\cap (e+Z_{\Lg}(f))\ne\emptyset$. The preimage $\pi^{-1}(U)$ intersects $\mu^{-1}(0)$. Thanks to smoothness of $\mu$, there exists a neighborhood $U_{Z(\Ll)}$ of $0\in Z(\Ll)$ such that $\pi^{-1}(U)$ intersects $\mu^{-1}(\zeta)$ for any $\zeta\in U_{Z(\Ll)}$. For $\zeta\in U_{Z(\Ll)}\cap Z(\Ll)^{\text{$(G/L)-reg$}}$ this means $U$ intersects $\Ad(G)(\zeta+X_n)$, and thus the asserted property (1) is true for ($X$ replaced by) $\zeta+X_n$.

Since nilpotent orbits are stable under scalar scaling, that (1) is true for $\zeta+X_n$ implies that it is true for $c(\zeta+X_n)$ for all $c\in\mathbb{C}^{\times}$ and $\zeta\in U_{Z(\Ll)}\cap Z(\Ll)^{\text{$(G/L)-reg$}}$, and thus also for $c\zeta+X_n$ since $c(\zeta+X_n)$ and $c\zeta+X_n$ are also conjugate. Since such $c\zeta$ covers all of $Z(\Ll)^{\text{$(G/L)-reg$}}$, the assertion for $X=X_s+X_n$ is proved.

[LS79] *Lusztig, G.; Spaltenstein, N.*, **Induced unipotent classes**, J. Lond. Math. Soc., II. Ser. 19, 41-52 (1979). ZBL0407.20035.

Sadly, I can't recall why I needed this cute property ….

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