It is indeed the case that if the two $\psi_i$ are in respective spin sectors $s_i$ (i.e., that $S^2|\psi_i\rangle = s_i(s_i+1)|\psi_i\rangle$ for $i=1,2$), then $\langle \psi_1 | \sigma_n^x | \psi_2 \rangle$ can be non-zero only if $|s_1-s_2|=0$ or $1$. The key idea is directly from ACuriousMind's answer, which I encourage you to read first. What follows is just a lengthy expansion of that answer to explicitly handle the multiplicity of identical irreducible representations. I expect there is a much more elegant way to do this, so alternative answers are encourage.
For any Hilbert space $\mathcal{H}_{N} = (\mathcal{H}_{1})^{\otimes N} \cong (\mathbb{C}^2)^{\otimes N} \cong \mathbb{C}^{2^N}$ of $N$ spin-1/2 particles, we can use the structure theorem for finite-dimensional von Neumann algebras to decompose it as
\begin{align}
\mathcal{H}_{N} \cong \bigoplus_{s={0,1/2}}^{N/2}\left(\mathcal{S}^{(N)}_s \otimes \mathcal{E}^{(N)}_s\right)
\end{align}
where $\mathcal{S}^{(N)}_s \cong {\mathbb{C}}^{2s+1}$ is an irreducible representation of the SU(2) group generated by $\{S^x,S^y,S^z\}$ and is spanned by the $2s+1$ orthonormal vectors $|s,m\rangle$ for $-s\le m \le s$. The direct sum starts at $s=0$ or $s=1/2$, depending on whether $N$ is even or odd, respectively, and the dimension of $\mathcal{E}^{(N)}_s$ — call it $D(N,s)$ — is just the multiplicity associated with how many copies of the spin-$s$ irrep there are. Any state $|\psi\rangle \in \mathcal{H}_{N}$ can therefore be written as
\begin{align}
|\psi\rangle &= \sum_{s={0,1/2}}^{N/2} \sum_{m=-s}^{s} \sum_{e=1}^{D(N,s)} \alpha(s,m,e) |s:m:e\rangle
\end{align}
with $\alpha(s,m,e)\in \mathbb{C}$ and where, using an arbitrary orthonormal basis $\{|s;e\rangle\}$ of $\mathcal{E}^{(N)}_{s}$, we have defined the orthonormal basis
\begin{align}
|s:m:e\rangle := |s,m\rangle\otimes|s;e\rangle \in \mathcal{S}^{(N)}_{s} \otimes \mathcal{E}^{(N)}_{s}.
\end{align}
For each fixed value of $s$ and $e$, and for $-s\le m \le s$, this spans a distinct spin-$s$ irrep of SU(2) in $\mathcal{H}_N$. Our assumption that the two $|\psi_i\rangle$ live in sectors with a fixed value $s=s_i$ means that, for them, the respective coefficients $\alpha_i(s,m,e)$ vanish except when $s=s_i$, i.e., the sum over $s$ becomes trivial.
We can alternatively decompose any $|\psi\rangle\in\mathcal{H}_N$ with respect to the tensor structure $\mathcal{H}_{N}= \mathcal{H}_{N-1}\otimes \mathcal{H}_{1}$ to get
\begin{align}
|\psi\rangle &= \sum_{\bar{s}={0,1/2}}^{(N-1)/2} \sum_{\bar{m}=-\bar{s}}^{\bar{s}} \sum_{f=1}^{D(N-1,\bar{s})} \sum_{\hat{m}=-1/2}^{1/2} \beta(\bar{s},\bar{m},f,\hat{m}) |\bar{s}:\bar{m}:f\rangle_{N-1} \otimes |\hat{m}\rangle_1
\end{align}
where $\beta(\bar{s},\bar{m},f,\hat{m}) \in \mathbb{C}$. Here we have used the structure theorem on $\mathcal{H}_{N-1}$ but not $\mathcal{H}_{1}$ since it is trivial for the latter. Parameters associated with $\mathcal{H}_{N-1}$ and $\mathcal{H}_{1}$ are indicated with bars ($\bar{s}$ and $\bar{m}$) and hats ($\hat{m}$) respectively. Analogously to before, we have defined the orthonormal basis
\begin{align}
|\bar{s}:\bar{m}:f\rangle_{N-1} := |\bar{s},\bar{m}\rangle\otimes|\bar{s};f\rangle \in \mathcal{S}^{(N-1)}_{\bar{s}} \otimes \mathcal{E}^{(N-1)}_{\bar{s}},
\end{align}
which, for each fixed value of $\bar{s}$ and $f$, and for $-\bar{s}\le \bar{m} \le \bar{s}$, spans a spin-$\bar{s}$ irrep of SU(2) in $\mathcal{H}_{N-1}$. This can be combined with the orthonormal basis $|\hat{m}\rangle \in \mathcal{H}_1$ for $\hat{m} \in\{-1/2,1/2\}$, which (trivially) spans the lone spin-$1/2$ irrep of SU(2) in $\mathcal{H}_{1}$,
to form a tensor-product basis for the joint Hilbert space of the pair of spins. In particular, we can perform the Clebsch–Gordan decomposition
\begin{align}
|s:m:\bar{s}:f\rangle:= \sum_{\bar{m}=-\bar{s}}^{\bar{s}} \sum_{\hat{m}=-1/2}^{1/2} C^{s,m}_{\bar{s},\bar{m};1/2,\hat{m}}|\bar{s}:\bar{m}:f\rangle_{N-1}\otimes|\hat{m}\rangle_1
\end{align}
with each allowed fixed choice of $s$, $\bar{s}$, and $f$ spanning a distinct spin-$s$ irrep of SU(2) in $\mathcal{H}_N$ for $-s \le m \le s$.
Recall that in our first decomposition of $|\psi\rangle$ we chose an arbitrary orthonormal basis $\{|s;e\rangle\}$ for $\mathcal{E}^{(N)}_{s}$; all that has happened is that, by fixing a value of $\bar{s}$ and $f$ in the state above, we have effectively picked out one of the basis vectors. In other words, there is a choice of basis for $\mathcal{E}^{(N)}_{s}$ and a function $e(\bar{s},f)$ such that $|s:m:e(\bar{s},f)\rangle=|s:m:\bar{s}:f\rangle$.
We then have
\begin{align}
\beta(\bar{s},\bar{m},f,\hat{m}) &= \Big[{}_{N-1} \langle\bar{s}:\bar{m}:f|\otimes{}_{1}\langle\hat{m}|\Big]|\psi\rangle\\
&= \sum_{s={0,1/2}}^{N/2} \sum_{m=-s}^{s} \sum_{e=1}^{D(N,s)} \alpha(s,m,e) \Big[{}_{N-1} \langle \bar{s}:\bar{m}:f | \otimes {}_{1}\langle\hat{m}|\Big] |s:m:e\rangle\\
&= \sum_{s={0,1/2}}^{N/2} \sum_{m=-s}^{s} \alpha(s,m,e(\bar{s},f))
C^{s,m}_{\bar{s},\bar{m};1/2,\hat{m}}
\end{align}
where to get the first (second) equality we used our first (second) decompositions of $|\psi\rangle$. To get the third equality we expanded in terms of Clebsch–Gordan coefficients and used the mapping $e\to e(\bar{s},f)$. Now, following ACuriousMind, we just make the key observation that the Clebsch–Gordan coefficient $C^{s,m}_{\bar{s},\bar{m};1/2,\hat{m}}$ vanishes unless $\bar{s}= s\pm 1/2$. This means that for $|\psi_i\rangle$, for which $\alpha_i(s,m,e)$ vanishes unless $s=s_i$, we can conclude that $\beta_i(\bar{s},\bar{m},f,\hat{m})$ vanishes unless $\bar{s} = s_i \pm 1/2$.
Finally, using the second decomposition for $|\psi_1\rangle$ and $|\psi_2\rangle$, we see that for any local operator on the $N$th qubit, say $\sigma_N^x$, we have
\begin{align}
\langle\psi_1|\sigma_N^x|\psi_2\rangle
&= \sum_{\bar{s}={0,1/2}}^{(N-1)/2} \sum_{\bar{m}=-\bar{s}}^{\bar{s}} \sum_{f=1}^{D(N-1,\bar{s})} \sum_{\hat{m}_1=-1/2}^{1/2} \sum_{\hat{m}_2=-1/2}^{1/2} \beta_1(\bar{s},\bar{m},f,\hat{m}_1)^* \beta_2(\bar{s},\bar{m},f,\hat{m}_2) {}_1 \langle \hat{m}_1|\sigma_N^x |\hat{m}_2\rangle_1
\end{align}
where we have used the orthonormality of $|\bar{s}:\bar{m}:f\rangle_{N-1}$. Having observed that $\beta_i(\bar{s},\bar{m},f,\hat{m})$ vanishes unless $\bar{s}=s_i \pm 1/2$, all terms in the above sum vanish except when $s_1 = s_2-1$, $s_2$, or $s_2+1$.
