Strictly, the argument I made about potential being constant inside a conductor because the charges arrange themselves to cancel the field inside is true only in a static situation. In a dynamic situation involving moving charges, such as current in a wire, this condition is violated. The actual charge distribution will be determined by the solution to Maxwell's equations for the given situation, and may include moving charges in the bulk (a good description for DC current) or more along the edges (better description for AC current)
Nevertheless, it is a very useful approximation to consider the potential as constant everywhere inside a conductor, even for a current-carrying wire.