Are there any particular techniques or products people use for damping the ringing strings behind the bridge on 330 or 360?
All guitars with a tailpiece will produce that ringing behind the bridge. Good thing about a Ric is that the ringing is easily blunted with no change to the look of the guitar.
On some guitars I just stuff a polishing cloth under those strings behind the bridge. Looks terrible. A cool way of doing it is to weave a dollar bill through the strings. Looks cooler than a polishing cloth.
With a Ric it is easy and neat. Just cut yourself a strip of black felt about three inches long and a quarter inch wide. Weave that between the strings and slide it down to the top of R tailpiece. The felt will fit underneath the front of the R tailpiece and be completely out of sight. That annoying ringing is now gone and the guitar looks the same.