How to conceptualize the action potential?
First of all, electric current is defined as movement of charges, $I=\frac{dQ}{dt}$. In electronics that you see around that is not very useful definition, because electrons move much slower than signals, that is changing electric field. Speed of electric field propagation reaches speed of light, whereas electrons move at 1%-30% or $c$.

In biology, especially regarding action potential, only movement of charges is important, not propagation of electric field, probably because conductance is much smaller than in copper wires. that is signal propagates not through interaction between charges, but via intermediate agent/amplifier -- ion channels.

Charge in body is more or less constant, because action potentials are cyclic: after depolarization membrane comes back to -70mV or something. There is no flow of charge out of the body.