Your initial statement, that $X$ admits a complex structure if and only if $N_J=0$ is not correct. I think you mean that $J$ comes from a complex structure if and only if $N_J=0$; that is correct, but in principle $X$ could be a complex manifold and $J$ a non-integrable complex structure.

But that's just a minor mis-statement; I know it's not what you meant. So let me make a suggestion that could be more to the point (though I don't have an answer to your question).

The vanishing of $N_J$ is equivalent to saying that the associated $\bar \partial$ operator satisfies $\bar \partial^2 = 0$. This is a condition that gives the existence of a lot of holomorphic functions in a neighborhood (under sufficient regularity assumptions). When $\bar \partial ^2 \neq 0$, the conditions for finding local holomorphic functions is too overdetermined, and in general you will have no holomorphic functions at all. Constructing these holomorphic functions, when $\bar \partial ^2 =0$, is a PDE problem called the Newlander-Nirenberg theorem. Perhaps if you follow the proof of that theorem, you could tell if the vanishing of $\bar \partial ^2$ (or maybe $N_J$) along certain directions related to your divisor could allow you to construct the functions you seek.

There is yet another equivalent statement to integrability; it is that the following: The almost complex structure $J$ has eigenvalues $\pm \sqrt{-1}$ so the complexification $T_X \times \mathbb{C}$ of $T_X$ splits as a direct sum $T^{1,0}_X\oplus T^{0,1}_X$ of eigenvectors with eigenvalue $\sqrt{-1}$ and $-\sqrt{-1}$ respectively. Integrability means that $T^{1,0}_X$ is closed under (the complexification of the) Lie bracket. If you assume the underlying structure is real analytic, then this integrability condition can be used with the complex version of Frobenius' integrability condition to give you (locally) a complex manifold in the complexified tangent bundle that projects onto the base. This gives the complex structure of $X$. Perhaps this picture could help you construct the divisor from holomorphic functions in the real analytic case; then if you get a good answer in this case, you can follow the method of Newlander-Nirenberg to adapt things in the smooth case.