No, old coins remained in circulation. This is one of the first lessons one learns in archaeology as to why a single data point cannot reliably be used to give a precise date--a coin of Trajan might be in a Trajanic context, or it could be in any later context.

Also, despite certain variability in metal content, bronze and silver coins generally circulated based on face value rather than metal value--gold coins probably did not, as they were rather more valuable.

Followup question: how did the Romans introduce newly minted currency to the economy? What was their federal reserve bank like? Or did the state just use newly minted currency to just buy stuff and hire people and get it into circulation like that?