The units for current density don't check out

In my lectures the equation for current density was given as:

[itex]J=n*q*vd[/itex]

where:
n = the number of electrons per unit area
q = electron charge
vd = drift velocity of electrons

If you rewrite this equation as:

[itex]J=\frac{N}{A}*q*\frac{dr}{dt}[/itex]

where:

[itex]\frac{N}{A}[/itex] is the number of electrons per unit area

and

[itex]\frac{dr}{dt}[/itex] is the distance each electron moves per second. ie. The drift velocity.
(this distance ignores the distance the electrons move due to their random thermal velocities. This is the distance that they all move together due to the electric field)

Now...

[itex]J= \frac{I}{A}[/itex]

[itex] \frac{I}{A}=\frac{N}{A}*q*\frac{dr}{dt}[/itex]

[itex] I =N* q*\frac{dr}{dt}[/itex]

N * q is the total charge

[itex] I =Q*\frac{dr}{dt}[/itex]

So the units for electrical current are coulomb meters per second.

However I know this is not true. The units of current are coulombs per second.

In the actual equation for current why is there no distance involved? I understand that you can model each coulomb as a particle, and then the current is a flow rate, ie. the number of particles per second. But when you try and derive this from current density it doesn't work out.

Will someone please explain how you can get from the current density equation, to the actual equation for current?

(the "actual equation for current" being [itex]I=\frac{dQ}{dt}[/itex])

Thanks!