interconverting Van Der Waal's equation for real gases from number of molecules to density
Your analysis is correct in terms of number density.  But let's see how it plays out in terms of molar density.
Let n be the number of moles and A be Avagadro's number.  Then N=nA.  If we substitute this into your first equation, then I get $$P=\frac{n(Ak)T}{V-nAb}-\frac{aA^2n^2}{V^2}$$But, since Ak=R, we obtain:$$P=\frac{nRT}{V-nb'}-\frac{a'n^2}{V^2}$$where $a'=aA^2$ $b'=bA$.  In terms of molar density $\rho=n/A$, this becomes:$$P=\frac{\rho RT}{1-\rho b'}-a'\rho^2$$