As long as you are staying within the same general platform (PC vs Mac) you do not need to change licensing. You are allowed to have two activated installations with your license, so you should be able to install on the new machine without having to worry about the older one at all (unless you ever want to free up the second activation for yet another machine). As you mentioned, you can download the trial version and use the serial number from your purchase to activate it to full use.
If you registered your software with Adobe, the serial number should be available via your Adobe account online. If you didn't register it then you will at least understand why it pays to do so.
IF you end up needing to resolve any issues with activating the product on the new machine you will need to contact Adobe Support directly. Here are some links to help make contact:
http://www.adobe.com/support/chat/ivrchat.html
http://www.adobe.com/support/download-install/supportinfo/