Magnetism, Radiation, and Relativity (1999)

Del[J] = - d rho(r,t) / d t

(those are partial derivatives). Basically that says that the rate of increase of the charge density inside an infinitesimal box is related to the total current flowing into the box, i.e. if the charge in a box changes, it must have flowed in/out from just outside the box, i.e. continuity (assuming no sources/sinks). Both exist at the same time. In SR, you can look at the relativistic four-current.

Maybe you're thinking of how a current can flow in a wire, despite the wire having no charge. This is because, on our length scales, the electric field of the positively charged atomic nuclei completely cancels the electric field of the negatively charged electrons.

In your example of a single charge moving at velocity v, which passes through the origin of your coordinate system at time t, the current density at the origin is basically a Dirac delta function. In other words, there is zero current density through that point at all times other than at one instant (vt) where it would be infinite (although finite when integrated over time, basically how Dirac delta functions work).

Thinking about it in this way may help you unite the gap between a moving and stationary charge.

You can also think of a Dirac delta function as a finite but normalised Gaussian (or normal) distribution of charge, which gets rid of the discontinuities. When that makes intuitive sense, take the limit where the width of the distribution approaches zero (but remains normalised so the amplitude becomes infinite).