Darboux like theorem for non-degenerate 3-forms in 6-manifolds we know Darboux theorem for higher-symplectic geometry is not correct in general, 
but is there any Darboux like theorem for non-degenerate 3-forms in 6-manifolds?
 A: This depends on what you mean by 'Darboux-like'.  It is certainly not true that a closed nondegenerate 3-form on a 6-manifold is necessarily locally equivalent to one of the 'flat' models, so there is no direct analog of the Darboux' theorem in this case.  
As I remark in my article "Remarks on the geometry of almost complex $6$-manifolds" (Asian Journal of Mathematics 10 (2006), 561–606, also available on the arXiv as arXiv:math/0508428), the closed $3$-forms of elliptic type in dimension $6$ essentially depend on 4 arbitrary functions of 6 variables (modulo diffeomorphism).  
However, there is an analog if you are willing to consider something stronger: If $\phi\in\mathcal{A}^3_+(M^6)$ is a $3$-form of elliptic type on an oriented $6$-manifold $M$, then there is a unique $J(\phi)\in\mathcal{A}^3_+(M^6)$ with the property that the complex $3$-form $\Upsilon = \phi+i \ J(\phi)$ is decomposable and hence of type $(3,0)$ with respect to an almost complex structure $J_\phi$ on $M$ that induces the given orientation of $M$.  All this is algebra.  However, now, if one adds the hypothesis that $d\Upsilon = 0$, which is, of course, the same as $d\phi = d\bigl(J(\phi)\bigr)=0$, then one has that there always exist local complex functions $z^1,z^2,z^3$ such that $\Upsilon = dz^1\wedge dz^2\wedge dz^3$.  In particular, $\phi = \mathrm{Re}(dz^1\wedge dz^2\wedge dz^3)$, so this is a sort of Darboux-like theorem; it's just that you need more hypothesis than the closure of the original form.
There is a similar result for nondegenerate $3$-forms of hyperbolic type.  This is the case in which the form $\phi$ can be written locally as $\phi = \phi_+ + \phi_-$ where each of $\phi_\pm$ is decomposable while $\phi_+\wedge\phi_-\not=0$.  (These two summands are unique up to permutation.)  In this case, the 'Darboux-like' theorem is that $\phi$ can be put in normal form if and only if $d\phi_+=d\phi_-=0$, which is stronger than $d\phi=0$ (which is not sufficient by itself).
Finally, there is the case of nondegenerate $3$-forms of what might be called 'nilpotent type' (which is not a stable type in Hitchin's sense, but is nondegenerate in the sense described by the OP).  A $3$-form $\phi$ on $M^6$ is nondegenerate of nilpotent type if and only if each point lies in some open set $U$ on which there exists a coframing $\omega^1,\ldots,\omega^6$ for which
$$
\phi = \omega^4\wedge\omega^2\wedge\omega^3
      +\omega^5\wedge\omega^3\wedge\omega^1
      +\omega^6\wedge\omega^1\wedge\omega^2.
$$
For such a $\phi$, the conditions that it can locally be put in this form with $\omega^i = dx^i$ for some coordinates $x = (x^1,\ldots,x^6)$ consist of two things:  First, the condition $d\phi=0$, which is clearly necessary; second, the condition that $3$-plane field $D\subset TM$ defined by $\omega^1=\omega^2=\omega^3=0$ (which is well-defined by $\phi$) should be Frobenius.  It is not hard to show that these necessary conditions are also sufficient, so this is the 'Darboux-like' normal form theorem in this case.
Note the interesting fact that, in each of these three cases, the 'Darboux-like' conditions are all first order equations on $\phi$.  This does not continue in higher dimensions.  In dimension $7$, the two stable types of $3$-forms each have examples that are flat to first order but not flat to second order, so the 'Darboux-like' theorems in this case turn out to involve a mixture of first and second order conditions.
