First, I'll show that you can assume that $ws_nw'$ is a reduced product.

To see this, first note that if we multiply $s_n$ by a reduced expression for $w'$, the result will still be reduced, and this can then be extended to a reduced expression for the longest element by multiplying some reduced expression $s_{i_1} \cdots s_{i_k}$ on the left, where $k$ is $1 + l(w')$ less than the length of the longest element...although, as of yet, we don't know that none of the $i_p$ are equal to $n$.

But then $s_{i_1} \cdots s_{i_k} s_nw' = ws_nw'$ so that $s_{i_1} \cdots s_{i_k} = w$. By the nature of the braid relations, any simple reflection that appears in $s_{i_1} \cdots s_{i_k}$ must appear in any expression for $w$, so if $s_n$ appears in $s_{i_1} \cdots s_{i_k}$, then it must also appear in $w$, which is a contradiction.

So, if we have your expression $ws_nw'$, we then in fact have a reduced expression for the longest element that has only a single occurrence of $s_n$.

Now consider any positive root $\alpha$ that has a nonzero coefficient on the simple root $\alpha_n$ corresponding to $s_n$ and also satisfies $w_0 \alpha = -\alpha$.

Claim: $\alpha$ must go negative at the location of $s_n$ in our reduced expression.

Proof: As you apply the simple reflections in our reduced expression one by one to $\alpha$, the only simple reflection that can affect the coefficient of $\alpha_n$ is $s_n$, and so when you get to $s_n$, the simple reflection $s_n$ must negate the coefficient of $\alpha_n$ (for otherwise $w_0 \alpha$ could not be equal to $-\alpha$). (That is, if $s_n$ just nullified the coefficient on $\alpha_n$, because $s_n$ doesn't occur again, one could not end up at $-\alpha$ after all the simple reflections in the reduced expression for $w_0$ have been applied.)

In every case except for type A, there are at least two such $\alpha$...for instance, you can take the sum of the simple roots and the highest root. But they cannot both go negative at the occurrence of $s_n$ in our reduced expression for $w_0$!

Therefore, no such expression exists except in type A.

And then in type A, there's the expression 1 21 321 4321 ... n (n-1) (n-2) ... 321.