This question is inspired by abx's comment to my previous question [MO430933][1]. 

Let $X$ be a complex surface of general type, and denote by $$a \colon X \to \operatorname{Alb}(X)$$ the Albanese map of $X$. 

It is classically known that the image $a(X)$ can be singular: for instance, if $C$ is a hyperelliptic curve of genus $3$  and $X=\operatorname{Sym}^2(C)$, then the Albanese map $a \colon X \to \operatorname{Alb}(X)$ is birational and contracts the $(-2)$-curve $\Gamma \subset X$ corresponding to the hyperelliptic involution of $C$ to an ordinary double point of $a(X)$.

However, in this example, the presence of the rational curve $\Gamma$ on $X$ implies that $\Omega_X$ is not globally generated. Moreover, this kind of situation is the only one I know in which there are singularities in the Albanese image. So, let me ask the

> **Question:** What is an example of surface $X$ such that $\Omega_X$ is globally generated and $a(X)$ is singular?

Note that the global generation of $\Omega_X$ implies that the Albanese map is unramified, namely, a local immersion at each point $x \in X$.
 


  [1]: https://mathoverflow.net/questions/430933/ampleness-of-the-normal-bundle-to-the-albanese-image