Of course. Gold was gold. The new emperor generally did try to recall the coins and have them re-forged with his face--because that portrait on the coin was likely the only way many people would know what the emperor actually looked like--but the old coins retained their value.

As others have pointed out, yes they were, and in fact, hoard evidence shows that coins of the Republic, which of course didn't have an emperor on them, were in circulation for well into the imperial period as well. But there are a few instances of coins of specific emperors being recalled and melted down due to Damnatio Memoriae. In particular, Dio Cassius records this happening to Caligula.