Let $X$ be a projective Calabi-Yau threefold with a single ordinary double point at $x \in X$, and smooth elsewhere. Is $X$ necessarily factorial?

I suspect that the answer is "yes", for the following reason. Some neighbourhood of $x$ is analytically isomorphic to the 'conifold' geometry $$ xy - wz = 0 ~~\mathrm{in}~~\mathbb{C}^4~. $$ This is not factorial; $x = w = 0$ is a non-Cartier divisor, and blowing up along this gives a small projective resolution of the conifold. However, analytic isomorphisms do not preserve the local class group, and $\mathcal{O}_{X,x}$ may nevertheless be a UFD. I suspect that this is necessarily true, since if not, we could blow up along the non-Cartier divisor, but one-nodal projective Calabi-Yau threefolds do not admit projective resolutions. I would like to know whether this argument can be made rigorous, or replaced by a simpler one.

Edit: It may be important to assume that $X$ is smoothable, and this is the case which matters to me.