Glad that tube fixed your problem!!
As for the background noise... is it the same level regardless of how loud the amp is set? Or does it go up and down in relation to the master volume?
It could be a number of things. The phase invertor tube might be going bad (since the noise is in all three channels). Could be related to the cable... is it still there even when the guitar volume is turned down? Could be in the guitar's wiring, possibly a loose ground connection or the volume pot could be getting worn.
Does it get better when you touch the strings? Does it change at all when moving the cable (crackling noise) or when you move the guitar between being perpendicular and being parallel to the amp? Is there any static or crackling when you turn either of the pots on the guitar?
Finally, all amps are going to have some background noise, so there may not be a problem at all. However, my first guess would be that you may have a dirty connection somewhere or a bad ground in either (or both) the guitar (I'd suspect the volume pot or the jack, but could be in several places) or in the cable. If you have other cables and/or guitars, you could see if it gets better or worse when you use a different cable or guitar.
Before investing in new cables or rewiring your guitar, I'd highly recommend getting some Deoxit contact cleaner and cleaning the pots and jack on the guitar. Wouldn't hurt cleaning the jack on the amp and the tube sockets, too. Deoxit has saved my butt more times than I can count (or care to remember).
FWIW, I use Mogami cables (among others) and have a PRS CU24. Pretty close to your setup.
Great guitar and a great amp!!