Moving lightroom to a new computer- Serial number trouble

The two options that I've read for finding your serial code don't seem to work in my situation.

We don't know what two options you are referring to.

However, if the old version is still installed you should be able to find the serial number in the Preference File: Preference file and other file locations | Lightroom Classic CC and Lightroom 6

As a general statement of good practices, you should always have some sort of record of ALL of your serial numbers (not just Lightroom). You need to find a method of keeping track of these, perhaps a text file, that is backed up to "the cloud". You should never be in the situation where you don't know the serial number(s) of your software.