Deriving goldman equation
The mistake is in the charge $q$, that you have put in:

$
D_A=\frac{\mu_{q}k_BT}{q}
$

$q$ is simply the particle charge (and not the whole charge, as you've unfortunately considered) and should then be expressed as:

$q=\frac{zF}{N_{A}}$

with $F$ the Faraday constant (charge / mole ), ${N_{A}}$ the Avogadro's number (number of particle / mole) and $z$ the charge valence.

${N_{A}}$ will finally cancel out when you divide Boltzmann's constant $k_B$ by the the charge $q$, in the definition of diffusion coefficient. That will allow you to recover the Goldman equation found on Wikipedia.

By checking all that through an always useful dimension analysis for $D_A$ and using the defintions you have given in your question for ($\mu_q$ = $\frac{v_d}{E} = \frac{v_d L}{V_m}$ and $k_B = \frac{R}{N_A}$):

$
\frac{\mu_{q}k_BT}{q}=\frac{m/s\,\times m}{V}\times \frac{J/(mol\times K)\,\times K}{{N_A}\times C/mol}\times{N_A}=\frac{m^2}{s}
$

you find the correct units for a diffusion coefficient.