Did Roger Chillingworth change his name to Roger Chillingworhth?I wrote in an essay that he did because it says "old Roger Chillingworth, as he was...

Yes, Roger Chillingworth did assume a false identity.  Prior to doing so, his last name had been Prynne.  Had Chillingworth been his real name, Hester would not have been Hester Prynne, but Hester Chillingworth. 

There were a few reasons for his choosing to go by a false name.  For one thing, Hester's affair and the resulting birth of Pearl, an illegitimate child, would have been cause for a great deal of humiliation directed at Chillingworth had anyone known his true identity.  In addition, Chillingworth's alias allowed him to infiltrate society and pursue his enemy without being detected by the man.  Because of his fake identity, Chillingworth was able to play the role of an innocent citizen and make others see him as he chose.