It's wise to execute a DPOA in the new state, for practical reasons more than legal ones. The idea is to have a document that banks and other institutions in that state are familiar with and that will be accepted. Institutions are familiar with the forms in that state, and some states have their own statutory forms. It's better to do a new POA than to risk having one from another state not accepted because it looks unfamiliar or doesn't contain the customary wording.
For more about the durable power of attorney, click here.