This follows from a result of Hopf (see the [exposition of Epstein][1]). By this result, one may assume that there is a disk $D\subset N$ such that $g^{-1}(D)$ is $d$ disks mapping diffeomorphically to $D$, where $d$ is the degree of the map (we may assume $d$ is positive by switching orientation if needed). 

Claim: Let $D\subset N$ be a ball. Then there is a map $h: N\to N$ such that $det(dh_x)=0$ for all $x\in N-D$ and $det(h_x)\geq 0$ for all $x$.

To prove this, cover $N$ with smooth balls $D_1,\ldots, D_k=D$ so that the interiors cover. Choose slightly smaller balls $B_i\subset D_i$ so that $B_i$ cover as well. We have maps $g_i: N\to N$ so that ${g_i}_{|N-D_i}=Id$, $det((dg_i)_x)=0$ for all $x\in B_i$, and $det((dg_i)_x)\geq 0$ for all $x\in N$ (this may be seen by using mollifiers on a ball). Then the composition $h=g_{k-1}\circ \cdots \circ g_1$ will have $det(dh_x)\geq 0$ for all $x\in N$ and $det(dh_x)=0$ for $x\in N-D$. 

Then $h\circ g$ will have the desired property. 



  [1]: https://doi.org/10.1112/plms/s3-16.1.369