I work closely with Immigration systems of UAE on Software development side. Following applies to you. 

 - Once you get your new passport, then your passport number is changed, that means that UAE visa is not more valid for you.
 - There is no process in UAE immigration system to transfer the visit visa to new passport. However this can be done for residence/Empoyment visas (not applicable to you)
 - You can not get a new visa until the previous one is cancelled. Ask your visa sponsor to cancel the visa if you wish to obtain new visa before expiry of current visa.
 - Cancellation is not required if new visa is obtained after expiry of old visa