Is there a connected $T_2$-space $(X,\tau)$ with more than 1 point and with the following property?
Whenever $D\subseteq X$ is dense, $X\setminus D$ is not dense.
A topological space is called irresolvable if it is not the disjoint union of two dense subsets. So you are asking whether there is a connected, $T_2$, irresolvable space with more than one point. The answer is yes! You can find a proof, along with a few references to relevant literature, in
D. Anderson, "On connected irresolvable Hausdorff spaces," Proceedings of the AMS, 1965. link
There is such a thing as a submaximal topology, in which every dense subset is open. These obviously satisfy your condition.
Take any connected Hausdorff space $(X,\tau)$. Let $\mathscr F$ be an ultrafilter of $\tau$-dense sets. Let $\tau'$ be the topology generated by $\tau\cup \mathscr F$. Then $(X,\tau')$ is submaximal Hausdorff connected.