To change the Client Company associated to a Project, navigate to the Project Details page, then click the name of the Client Company currently associated to the Project. Doing so will bring up a drop-down menu of all available Client Companies, from which you can then choose the new Company:
To change the Client Contact associated to a Project, click the "Edit" symbol in the Project Contacts modal on the right-hand of the page, then scroll down to the available Client Contacts.