Let $X$ be a non-singular, quasi-projective variety (over $\mathbb{C}$) of dimension at least $3$, $D_1, D_2$ are integral effective divisors in $X$ with $D_1 \cap D_2$ of codimension $2$ in $X$. Let $C \subset D_1$ be an integral closed subscheme in $D_1$ of codimension $1$. Note that, as $C$ is irreducible and contained in $D_1$, we have $C \cap (D_2 \backslash D_1 \cap D_2)=\emptyset$. Does there exist an effective Cartier divisor $D \subset X$ containing $C$ such that $D \cap (D_2 \backslash D_1 \cap D_2)=\emptyset$ and $D \cap D_1$ is of codimension $2$ in $X$? Moreover, if $C$ is non-singular can we get such a $D$ which is also non-singular?

If necessary, assume that $D_1, D_2$ and $D_1 \cap D_2$ are non-singular as well.