Hausdorff spaces and continous functions on dense sets

The idea of going by contradiction is fine, but you need a small creative step (inspired by (b)) to abuse the power of assumption (c):

Suppose that $x$ and $y$ are distinct points that satisfy your $(1)$ or:

$$x \neq y \land \forall O,O' \subseteq X \text{ open }: (x \in O \land y \in O') \implies O \cap O \neq \emptyset$$

so we cannot separate $x$ from $y$ by open sets.

Now consider $X \times X=X^2$ in the product topology, let $Z=\overline{D}$, where $D=\Delta(X)=\{(x,x)\in X^2: x \in X\}$ in the subspace topology. Let $f_1(x,y)=x$, restricted to $Z$, and $f_2(x,y)=y$, restricted to $Z$, be the (continuous) projections.

By construction $D$ is dense in $Z$ and $f_1$ and $f_2$ are the identity restricted on $D$, so coincide. The assumption $(1)$ says that $(x,y) \in Z\setminus D$ and $f_1(x,y) \neq f_2(x,y)$, contradicting (c).