What is the process for moving the software from an old computer to a new computer. Do I have to deactivate the software on the old computer and then download again onto the new computer?
Which Adobe software exactly?
You can activate the software on a maximum of two computers at the same time so you could install on the new and leave the old as it is.
If you want to free up the activation on the old computer (say, to use the software elsewhere) then you must first deactivate it via Help > Deactivate.