Answer:
Most conductors are fully charge neutral when carrying current, since their atomic structure is not altered, and electrons are in their normal state of jumping from atom to atom all the time.
It is just that this movement gets a direction under an external electric field, which causes a net charge flow (current) in conductor. So the conductor remains uncharged.
It is however possible for a capacitor plate to be charged (charged capacitor), and at the same time the plate can be a part of current carrying circuit through two points on it. In this case, charge on plate is independent of current and unrelated with it.
I have made foil type capacitors with such construction where the capacitor was a part of filter circuit in a signal carrier circuit, and connections were made from both ends of each foil.
