Let $E$ be an elliptic curve, $A$ an abelian surface, and $x\in A$ a point that's not 2-torsion.

Let $Y \subset A \times E$ be a surface of high degree, stable under inversion, and containing $x \times E$, that is general among hypersurfaces satisfying these conditions. 

If the degree is sufficiently high then (by a dimension-counting argument) a general such $Y$ will be smooth and not contain any two-torsion points of $A \times E$.

So the quotient of $Y$ by inversion will be smooth. Call this $X$.

I claim $X$ is general type. Indeed, its unramified double cover $Y$ is an ample hypersurface in an abelian variety, hence of general type.

I claim $\Omega_X$ is not globally generated, and indeed has no global sections - they would pull back to involution-invariant global sections of $\Omega_Y$, which by Lefschetz would give involution-invariant global sections of $\Omega_{A \times E}$.

I claim $\Omega_X$ is not ample. Restricted to the elliptic curve $x \times E$, it has the rank one trivial quotient $\Omega_E$, and thus cannot be ample.

I claim $\operatorname{Sym}^n \Omega_X$ is globally generated for all even $n$. Indeed, every section of $\operatorname{Sym}^n ( H^0(A \times E, \Omega_{A \times E}))$ for even $n$ is involution-invariant and thus, restricted to $Y$, descends to $X$, and these sections locally generate $\operatorname{Sym}^n \Omega_{A \times E}$, hence they also locally generate $\operatorname{Sym}^n \Omega_Y$ and $\operatorname{Sym}^n \Omega_X$. 

So $X$ is an example.