If you registered your product at the Adobe web site, you can recover your serial number from there, see these instructions:
If that doesn't help, then I would normally suggest you contact Adobe Customer Care Live Chat, but it sounds like you have done that. We can't really help with serial number issues here in this forum.
And I also recommend that from now on you keep better track of your serial numbers (not just for Lightroom but for all products). I use a text file on my computer named "Serial Numbers.txt" and save a copy to cloud backup so I can always get to it.
Where is the previous installation/computer with the serial number??
I have sent you a personal message.
Please check and reply there.