There should be a serial number on the package if you installed from that.

Alternatively, sign on to Finale and look for your purchased products, which should lead you to the SN.

No, no S/N on the package. I have checked everywhere. This finale 2012 did not request a S/N when I bought it 4 years ago.

I signed on to Finale and no way can lead me to the S/N

You have a MakeMusic Users Account. If not, you wouldn't be able to post here. Your copy of Finale should be linked to this account as is your serial number. Being 2012 or later, you should be able to find, download and reinstall. If not, use the Submit a request link at the top of the page and request help directly from MakeMusic. You can use the Manage Authorizations button to de-authorize your old computer.

A couple of exceptions:

If your new computer is a Mac running OS 10.13 High Sierra, 2012 will not install. You should be able to upgrade to Finale 25 from within your account. The price will depend on the Finale version you currently have.

If your copy is part of an institution or education site license, you will probably have to contact that License Administrator.