Procedure for simulating nonlinear flight dynamics
I am not a flight dynamics expert, but let me give an extremely general answer on verification of a non-linear dynamic system.  consider that you have the non-linear system $\dot x =f(x)$.  

This system may have multiple equilibrium points, which are the $x$ values such that $f(x)=0$.  You have checked that one of the desired equilibrium points is actually an equilibrium point in your simulation. So that's good.  But does your system have more than one equilbrium point?  if so, you probably want to check all of them.

Next, you want to check the stability of the equilibrium points.  You've checked that $f(x)=0$ for one equilibrium point.  You need to check what happens for a small perturbation say $x_0= x+\epsilon$.  If the equilbrium point is supposed to be stable, then you need to check that after some time, the system has returned to equilibrium.  If the equilbrium point is supposed to be unstable, you need to check that the system has diverged from the equilbrium point.

But even that is not good enough.  You also need to check the rates of convergence or divergence.  i.e. near the equilbrium point, you can linearize the system and find the eigenvalues.  i.e. if you are at a small perturbation away from a stable equililbrium, you expect the system to approach equilibrium at a certain rate, say $x=x_0 e^{-\lambda t}$.  You need to verify that the simulation matches the expected rate.  

So to summarize, you need to verify that your simulation has the correct number of equilibrium points, that the stability of each equilbrium point is correct, and that the eigenvalues near each equilbrium point are correct.