Let $X$ be a minimal surface of general type. Recall that a vector bundle $\mathscr{E}$ on $X$ is called globally generated if the evaluation map of global sections $$e \colon H^0(X, \, \mathscr{E}) \otimes \mathcal{O}_X \to \mathscr{E}$$ is surjective. Instead, $\mathscr{E}$ is called $ample$ if the line bundle $\mathcal{O}_{\mathbb{P}(\mathscr{E})}(1)$ is ample on $\mathbb{P}(\mathscr{E})$. By the cohomological characterization of ampleness, if $\mathscr{E}$ is ample then the symmetric power $\operatorname{Sym}^n \mathscr{E}$ is globally generated (and ample) for $n \gg 0$.

Now let us take $\mathscr{E}=\Omega_X$, the cotangent bundle. I'm looking for examples of $X$ such that:

1. $\Omega_X$ is neither globally generated nor ample;
2. $\operatorname{Sym}^n \Omega_X$ is globally generated for $n \geq n_0$, where $n_0 \geq 2$ is an explicit constant (I would be already happy with $n_0=2$).

Note that these conditions provide several restrictions. For instance, 1. tells us that the Albanese map $a_X \colon X \to \operatorname{Alb}(X)$ is not a local immersion, since the non-surjectivity of the evaluation map for $\Omega_X$ at a point $x \in X$ is equivalent to the non-injectivity of the differential $da_X(x)$. Moreover, 2. implies that $K_X$ is ample: indeed, by my previous question MO412306, if $\operatorname{Sym}^n \mathscr{E}$ is globally generated then $X$ contains no rational curves at all; as a consequence, the failure of $a_X$ to be locally immersive is not related to the contraction of rational curves.

I have looked for such examples, so far without success. Probably, one of the reasons is that I do not know a geometrical characterization of the global generation of $\operatorname{Sym}^n \Omega_X$ in terms of the Albanese map. So let me ask the following

Question. What are examples of minimal surfaces of general type that satisfy 1. and 2. above? More generally, how can I check in general whether $\operatorname{Sym}^n \Omega_X$ is globally generated?

Edit. Will Sawin's nice answers provide examples $X$ which are étale quotients $Y \to X$, where $Y$ is a suitable hypersurface in a reducible abelian threefold. Then $X$ has zero second Segre class, namely, $s_2(X)=c_1(X)^2-c_2(X)=0$, because $Y$ has the same property and this is preserved by étale maps.

Question 2. Can we obtain examples with positive $s_2$?

My guess is that this should be possible, by generalizing Will's construction to étale quotients of suitable complete intersections in reducible Abelian fourfolds.