Tampering with Shamir's subshares
Let $J$ be the set of conspirators and $S$ the total set of shareholders.

In general you are correct, provided the polynomial $A'(x)-A(x)$ is not zero at $x=0$, where

$$A'(x)=\sum_{j \in J} y_j \prod_{k \in J:k \neq j} 
\frac{ (x+W- x_k)}{(x_j+W-x_k)}
\prod_{k \in S \setminus J} 
\frac{ (x-x_k)}{(x_j-x_k)},
$$ 
and
$$A(x)=\sum_{j \in J} y_j \prod_{k \in J:k \neq j} 
\frac{ (x- x_k)}{(x_j-x_k)}
\prod_{k \in S \setminus J} 
\frac{ (x-x_k)}{(x_j-x_k)}.
$$ 

The equation can be simplified into $$A'(0)-A(0)=0$$ by letting $x=0,$ into a polynomial in $W$ whose roots you have to avoid if you want to obscure the secret from the innocent shareholders.

Since the difference is a Lagrange polynomial  it is as secure as ordinary secret sharing so if you aren't in possession of $W$, its effect is uniformly distributed over the finite field.