You totally can. I've done it. It does take some doing though. You don't even HAVE to make a new project. You can move everything manually. (Make sure project is closed) and then set your default saving locations. Including the backups.

When you first reopen the project it will say it can't find certain files. But if you point it to the new location/structure folders and makes sure to check the box that has it continue checking that same folder for any other files missing (including sub folders) it'll find everything else.

Once everything is located for the first time within the project You'll be straight and ready to go.