I guess I'm looking for the dirty channel to be a bit...thicker...I hope that makes sense. The clarity on the dirty channel isn't bad; it doesn't sound muddy at all. The clean channel is good, but the dirty is just not quite what I'm looking for...
There are a few things you might try, though of course, one man's thick is another man's WTF...so take my advice with a grain of salt, please.
By coincidence, in addition to my PRS amps, I have a Mesa Lone Star, whose cleans I like when I want very spanky cleans, but whose OD channel is a bit hard to get that saturated but not over-the-top sound with. I like it very much with a Suhr pedal called the KokoBoost. It's simply a magic box. You don't have to push the front end of your amp with more volume. Even at unity gain, it's a "make it better sounding" box.
And you can use it in standard mode, where it just makes things...uh...a little thicker and more "there"
...or in mid boost mode, where you can select the frequency you want to emphasize. The pedal is one of my little secrets for recording.
Another thing you might try in the pedal department is a good EQ pedal, if for no other reason than to find out which frequencies you'd like to mess with. People underestimate what a graphic EQ pedal can accomplish, they'll rush out and buy a new amp when a bit of EQ in the right frequency band can solve the problem.
In the tube department, I'm not a big fan of the JJ 6L6 tubes for some reason. I do like the Tung Sol 6L6 GC reissues if you don't want to spring for NOS.
And of course various vendors have NOS, which usually is an improvement in tone both clean and dirty. I've seen Sylvania and RCA recently at not too scary prices. You won't find better 6L6GCs than those. The Winged C 6L6s are now considered NOS, but they're worth investigating for a bit less dough.
Preamp tubes can also make a big difference, and I'd at least try NOS in V1. I have NOS preamp tubes in both my HXDA and DG30, and they sound great, clean or dirty.
And of course speakers will make the biggest difference of all, but that's an encyclopedia in itself and you just need to listen to a bunch.
Good luck on your quest!