You can use your TurboTax Desktop code on up to 3 different computers, so you can just use your existing code, if you haven't already used it 3 times before.
TurboTax Desktop only saves your return files to your computer, not on our servers. So, you will have to copy the files from your old computer onto your new computer to be able to see them there. You can use a USB or other external drive to transfer the files.