Here is a construction of a family of such examples that will work, but you will have to choose a particular map to get an explicit example.
Let's use coordinates $v_1,v_2,v_3,v_4,x,y$ (each periodic of period $2\pi$). Choose a smooth map $u=u(x,y):\mathbb{T}^2\to S^2$ that has nonzero degree. (This will ensure that it is not null-homotopic.) One possible example would be to make it a $2$-to-$1$ branched conformal double cover (i.e., it will have 4 branch points), the classical way to exhibit an elliptic curve as a branched double cover of $S^2=\mathbb{CP}^1$.
Write $u = (u_1,u_2,u_3)$ and consider the following $2$-form on $\mathbb{T}^6$
$$
\begin{aligned}
\omega &= u_1\,(\mathrm{d}v_1\wedge\mathrm{d}v_2+\mathrm{d}v_3\wedge\mathrm{d}v_4)
+u_2\,(\mathrm{d}v_2\wedge\mathrm{d}v_3+\mathrm{d}v_1\wedge\mathrm{d}v_4)\\
&\qquad +u_3\,(\mathrm{d}v_3\wedge\mathrm{d}v_1+\mathrm{d}v_2\wedge\mathrm{d}v_4)
+ \mathrm{d}x\wedge\mathrm{d}y
\end{aligned}
$$
This $2$-form is compatible with the flat metric $g = {\mathrm{d}v_1}^2 + {\mathrm{d}v_2}^2 + {\mathrm{d}v_3}^2 + {\mathrm{d}v_4}^2 + {\mathrm{d}x}^2 + {\mathrm{d}y}^2$ and hence defines a $g$-orthogonal almost complex structure on $\mathbb{T}^6$ by the usual rule $\omega(w_1,w_2) = g(Jw_1,w_2)$ for all tangent vectors $w_1,w_2$.
I think this $J$ is probably not integrable for any nonconstant $u$ (and hence for $u:\mathbb{T}^2\to S^2$ of nonzero degree), but it will certainly be nonintegrable for `generic' $u$. The nonzero degree requirement on $u$ implies that $c_1\not=0$ for this almost complex structure.
