The first thing you should do is look up on Adobe's site and see if you can find the serial number in the My Products section of your profile. If not, then try to register the software on the old computer before deactivating or uninstalling. That should transmit your serial number to Adobe and you can look it up at adobe.com in the My Products section of your Adobe profile.
If trying to register doesn't work, then my reading of that webpage suggests that when you uninstall and DON'T ERASE the serial number, when you reinstall on the same hard-drive, the serial number will be found automatically. The serial number is apparently stored encrypted somewhere on the hard-drive so if you install to the same hard-drive it'll be found, but if you to install on a new hard-drive that encrypted information would no longer be available so you'd have to type in the serial number.
So, to find your serial number using the method described on that webpage, uninstall without erasing the serial number, then immediately reinstall to the same, old hard drive and the activation wizard should show you the serial number as it shows on that webpage. Once you see the serial number, write it down and/or copy-paste it into a document, and use it to install on the new hard-drive, after uninstalling from the old one, again.
You can generally install Adobe software on up to two computers simultaneously--like a work computer and a laptop that you might use while away from work. Because you can deactivate on one computer and reactivate on another, Adobe has also put a limit on the total number of activations for a serial number, to keep someone from using the software on many computers by deactivating and reactivating every time they use it somewhere else. At one point in the past Adobe limited things to 20 total activations, but nowadays, they probably just watch how you are using and activating the software and limit you if they feel you're trying to get around their single-user 2-computer licensing scheme, and the limit may not be exactly 20. If you do run into the 20 activation limit, you might be able to get them to reset that by calling or chatting with customer service, but they may tell you that your old version is no longer supported, or they may tell you No for some other reason.