Let's begin by reformulating the question a bit. Note that any orientable 4-manifold is spin-c, so in particular has $w_3 = 0$. The condition that $w_1 \not= 0 $ is thus redundant.

Wu's theorem gives that on a closed 4-manifold

- $w_1 = v_1$, so $Sq^1(c) = w_1 c$ for any $c \in H^3$ ($\mathbb{Z}/2$ coefficients everywhere)
- $w_2 = v_2 + v_1^2$, so $v_2 = w_2 + w_1^2$. Thus $(w_2 + w_1^2)z = z^2$ for any $z \in H^2$
- $w_3 = v_3 + Sq^1(v_2) + Sq^2(v_1)$. Because $Sq^2(v_1) = 0$ and $v_3 = 0$ for degree reasons, this implies $w_3 = Sq^1(w_2)$.

We also have universally that $w_3 = Sq^1(w_2) + w_1 w_2$, so for closed 4-manifolds we get $w_1 w_2 = 0$. Putting these together we find

$$w_3 x = Sq^1(w_2) x = Sq^1(w_2 x) + w_2 Sq^1(x) = w_1 w_2 x + w_2 x^2 = x^4$$

on any closed 4-manifold with $w_1^2 = 0$. Hence the question is equivalent to looking for a closed 4-manifold with $w_1^2 = 0$ and some $x \in H^1$ such that $x^4 = 1$.

We can now approach the problem by the following strategy. Look for closed 4-manifolds with the right characteristic numbers. If there are none, then the answer to the question is negative. If there is one, then try to obtain a solution to the problem by surgery.

In this case we need a closed 4-manifold with a distinguished class $x \in H^1$, and the relevant characteristic numbers are $x^4 = 1$, $w_1^4 = w_1^3x = w_1x^2 = 0$; actually one condition is redundant because $w_1^3 x = Sq^1(w_1^2 x) = w_1^2 x^2$.

$RP^2 \times RP^2$ has $w_1^4 = 0$, and if we take $x$ to be the generator from one of the two factors then $w_1^2 x^2 = 1$ while $x^4 = 0$. On the other hand $RP^4$ has $w_4 = 1$, and if we take $x$ to be the generator then $w_1^2 x^2 = x^4 = 1$, while if $x = 0$ then of course $w_1^2 x^2 = x^4 = 0$. Hence if we let $M$ be the connected sum of $RP^2 \times RP^2$ and two copies of $RP^4$ then $M$ is a closed 4-manifold with a class $x \in H^4(M)$ such that $w_1^4 = w_1^2 x^2 = 0$ and $x^4 = 1$.

$H^1(M) \cong H_1(M)$ has rank 4. Any $\alpha \in H_1(M)$ with $w_1(\alpha) = 0$ can be represented by an embedded circle with trivial normal bundle. Doing surgery on this circle cuts down the rank of $H^1$ by 1. If in addition $x(\alpha) = 0$, then the result of the surgery will still have a class $x$ with the desired properties $x^4 = 1$, $w_1^4 = w_1^2 x^2 = 0$. So by doing surgery twice on $M$ we obtain an $X$ with the right characteristic numbers, but $H^1(X)$ of rank 2 generated by the essential classes $w_1$ and $x$.

In particular $w_1^3$ and $w_1^2 x$ both pair to 0 with all elements of $H^1(X)$, so must vanish. Thus, if we let $S \subset N$ be an embedded smooth surface Poincare dual to $w_1^2$ then $w_1$ and $x$ both vanish on $S$. It would be nice if we could deduce that $S$ can be taken to be an embedded sphere so that it can be killed by surgery, but that's not obvious to me.

However, we can cut out a tubular neighbourhood of $S$ to obtain a 4-manifold $Y$ with boundary. Then $x \in H^1(X)$ has a preimage $y \in H^1(Y, \partial Y)$. Now glue $Y$ to itself along the boundary to obtain a closed 4-manifold $Z$, and let $z \in H^1(Z)$ be the image of $y$ from one side. Then $w_1(Z)^2 = 0$ and $z^4 = 1$, so $w_3(Z)z = 1$.