Just download CS5, install it from scratch on the new computer after deactivating it on the old one, then input your valid serial number when you go register it and activate right then.
You can use a FTP client (or your browser if it supports FTP) and download from the Adobe FTP server (ftp.adobe.com).
Make sure you have a full legal serial number. You cannot get it from the installation of the old computer, as the splash screen will show you only a DELIBERATELY TRUNCATED serial number. It's missing four digits to discourage software piracy.