I have a serial number...I put it in, during the install...it doesn't allow me to install without a valid s/n. Once installed, it says I'm running in trial mode and I have 60 days to purchase.
I'ver never seen that LiveCycle Designer has required an activation. All you need is the license key which looks this way: 1234-4567-8901-2345-6789-0123.
Just, reinstall Designer ES4 on the new PC with your existing serial number. You'll need your CD-ROM or previously downloaded installer to perform the installation.