There is a general fact that a proper affine morphism is finite, and a general trick that for $\overline{X} \to S$ proper containing an open subset $X\to S$ finiteaffine, a closed subscheme $Z$ of $\overline{X}$ which is in fact closed in $X$ is a closed subscheme of $\overline{X}$, hence proper, and also a closed subscheme of $X$, hence affine, and thus finite. Combined with the surjectivity you note this gives what you want.
Depending on the definition of affine curve, I think your argument with Weil divisors won't work. Consider $S$ a curve and $X$ a family of $\mathbb P^1$s that degenerates to two $\mathbb P^1$'s joined with a node over a single point. We can take one of the Weil divisors to be one of the $\mathbb P^1$s and the other to be a section passing through the other $\mathbb P^1$s. Then they don't intersect, the problem being that the complement of the section is not affine. (Concretely, work over the base $k[t]$ and take the family of curves in $\mathbb P^2$ with equation $xy-t z^2$ where one Weil divisor is the section $(0:1:0)$ and the other is the curve where $t=y=0$.)