If $[\Sigma]\in H_{n-1}(M,\mathbb Z)$ is primitive, its image $[\Sigma] \in H_{n-1}(M,\mathbb Z/2\mathbb Z)$ can always be represented by a non-orientable manifold! Since it is primitive, we may suppose that $\Sigma$ is connected, and since $[\Sigma]\neq 0$ the manifold $M\setminus \Sigma$ is connected and hence there is always an arc in $M$ connecting two points of $\Sigma$ from opposite sides. If you self-connect sum $\Sigma$ along this arc you get a non-orientable representative of the same homology class in $H_{n-1}(M,\mathbb Z/2\mathbb Z)$.

If $[\Sigma]\in H_{n}(M,\mathbb Z)$ is primitive, its image $[\Sigma] \in H_{n}(M,\mathbb Z/2\mathbb Z)$ can always be represented by a non-orientable manifold! Since it is primitive, we may suppose that $\Sigma$ is connected, and since $[\Sigma]\neq 0$ the manifold $M\setminus \Sigma$ is connected and hence there is always an arc in $M$ connecting two points of $\Sigma$ from opposite sides. If you self-connect sum $\Sigma$ along this arc you get a non-orientable representative of the same homology class in $H_{n}(M,\mathbb Z/2\mathbb Z)$.