Let $X$ be a connected separable metrizable topological space. Call it a cut-point space if $X\setminus \{x\}$ is disconnected for every $x\in X$. Then does $X$ embed into the plane?

My thoughts:

(0) It is not difficult to see that $X$ must have dimension $1$, and therefore embeds into $\mathbb R ^3$.

(1) Every separable cut-point space has only a countable set of cut points of order greater than 2 ($x$ is a cut point of order $n\in \{0,1,2,...\}\cup \{\infty\}$ if $X\setminus \{x\}$ has $n$ components). Coincidentally, or perhaps not, there is no uncountable collection of disjoint "tacks" in the plane: How many tacks fit in the plane?.

(2) Every dendrite embeds into the plane. A dendrite is a locally connected continuum $X$ such that for every two points $x,y\in X$, there is a third point $z\in X$ such that $x$ and $y$ belong to different components of $X\setminus \{z\}$. It is not clear to me whether every cut-point space can be embedded into a dendrite.