Shimada proved this result in Singularities of Dual Varieties in Characteristic 3 in 2006, by assuming the linear series is *"sufficiently ample"*:

**Theorem**: *Let $X$ be a smooth projective variety of dimension $n>0$, over an algebraically closed field $k$ with $\text{char} \ k>3$ or $\text{char}\ k=0$. Take a sufficiently ample embedding $|\mathcal{L}|:X\hookrightarrow\mathbb P^N$, then a general plane in $(\mathbb P^N)^{\vee}$ cuts the dual variety along a curve with only ordinary cusps as its unibranched singularities.*

Note that by picking the plane general, the "multibranched" point should have exactly two branches, and such singularities are just ordinary nodes.

Here, the "sufficient ampleness" of a linear system $|\mathcal{L}|$ is the surjectivity of the evaluation map

$$v_p:\mathcal{L}\to\mathcal{L}_p/\mathfrak{m}_p^4\mathcal{L}_p\cong \mathcal{O}_p/\mathfrak{m}_p^4 \tag{1}$$

for all $p\in X$. This condition gurantees that up to the third order tangent cone is reachable by restricting linear functionals from the ambient projective space to a neighborhood at each point.

To get some understanding, we put $\mathcal{C}:=\{(p,H)\in X\times(\mathbb P^N)^{\vee}|\ p\in X\cap H \ \text{is singular}\}$. The image of the second projection

$$\pi:\mathcal{C}\to (\mathbb P^N)^{\vee}\tag{2}$$

is called the *dual variety* and denoted as $X^{\vee}$, which has expected dimension $N-1$. Therefore, a general plane cuts $X^{\vee}$ along a curve $\Sigma$ whose singularities are general in $(X^{\vee})^{sing}$. We know that a smooth point of $\Sigma$ corresponds to a hyperplane section with a single ordinary node. But assuming $|\mathcal{L}|$ being sufficiently ample $(1)$, if $q$ is a unibranched singularity of $\Sigma$, then it corresponds to a hyperplane section with a single $A_2$ singularity (analytically equivalent to $x_1^2+...+x_{n-2}^2+x_{n-1}^3=0$ working over $\mathbb C$). More precisely, if we put
$$\mathcal{E}^{A_2}:=\{(p,H)\in X\times(\mathbb P^N)^{\vee}|\ p\in X\cap H \ \text{is singular of type $A_2$}\},$$

the author showed in Prop. 4.9 that $\mathcal{E}^{A_2}$ is Zariski dense in $\mathcal{E}$, which has codimension one in $\mathcal{C}$ and is defined as the singular locus of $(2)$. Finally, in Theorem 5.2, the author showed that such $q$ is an ordinary cusp.

This gives a sufficient condition to solve @IMeasy's question, but I'm not sure if we can remove "sufficient ampleness" condition in $\text{char}\ k=0$ or even over $\mathbb C$. Perhaps, one should study examples which violate the conclusion of Prop. 4.9.