If by "subvariety" you mean that $W$ is known to be irreducible
Let $Z$ be the base locus of $D$. The sections give a map $T \setminus Z \to \mathbb P^{11}$. Your claim about six general sections implies that $W$ does not intersect $Z$ (otherwise the intersecion with arbitrarily many sections would include $W \cap Z$ and be nonempty) and that the image of $W$ along this map has dimension at most $5$ (otherwise the intersection of this image with $6$ general hyperplanes would be nonempty). The claim about five general sections implies the image contains a $5$-dimensional component (otherwise the intersection of this image with five general hyperplanes would be empty) and that the fiber over a general point of this $5$-dimensional component is zero-dimensional (otherwise the intersection with the inverse images of five general hyperplanes would include the positive-dimensional fiber).
Certainly $W$ contains an irreducible component which maps dominantly to the $5$-dimensional component of the image of $W$ and hence is $5$-dimensional. However, if $W$ is not irreducible, there's no way to rule out that $W$ contains also a larger-dimensional component with smaller-dimensional image in $\mathbb P^{11}$.