Considering Young Tableaux filled with numbers $1,...,n$ in a natural way, (left-to-right, row-by-row), the book "Representation Theory" of Fulton and Harris (Exercise 4.24) states that for all $x$ in $\mathbb{C}S_n$,
$c_\lambda x c_\mu = 0$, if $\lambda \neq \mu$.
However, they same is also true when considering tableaux $A$ of shape $\lambda$ and $B$ of shape $\mu$ filled in any other manner (with no repetitions) with the numbers $1$ to $n$. That is, for all $x$ in $\mathbb{C} S_n$,
$c_{\lambda,A} x c_{\mu,B} = 0$, if $\lambda \neq \mu$.
Fulton and Harris do not consider this generalisation; and the book "Symmetric Group" of Sagan is using Specht Modules.
What would be a a suitable reference for the above statement?
Actually, book proves it for any Tableaux. The chapter starts with "We will assume we take the Standard Tableaux" (The Tableaux you mentioned) but then ignores completely that and does It in the general case. In fact, in many of the Lemmas its important for the Tableux to be arbitrary since many times he builts new Tableaux and applies propositions to those Tableaux.