Your "muddy" description is why I play clean almost all of the time - I have had no luck yet to set up my equipment to give me a satisfying sound when distorted/overdriven/fuzzed - things turn into a mess. Frustrating. I want string definition, clarity, not noise. It is the reason my plans to build a pedal board and decent noise-free power supply without daisy chains has not progressed beyond plans the past three or more years. And, the one pedal refuses to work with any power supply other than it's brand name...
With "Toy" I refer to playing for fun, which, when it works out, the toy becomes a tool. Someone once said to me: "A rifle is a tool. A tool needs to do a specific job. Alter it to do that job."
OK, so getting past the feel guilty ("ouch!") for cutting up and modifying (and devaluing) expensive equipment, "toy" as well as "tool" needs to work. Thus the endless modifying, setting up, experimenting, and constant fiddling to have fun when playing (or success when working).
I have no knowledge of active pickups, those you have now could be labelled "active passives" if I understand the design parameters? (You will have googled a lot already, pap batteries are blamed, string height is blamed, too much output is blamed, going to 18V is advised, replacing a crappy amplifier seems to be common advice. Sure. Replacing a crappy guitar as well. Sure. Ignore those.) Someone with experience can tell us if going to larger value pots might help brighten the tone?
Hope you get the Cort's playing setup sorted for you, and then play around with the EMG pickups to see if you can get it closer to what you think you want. You might get a "better" guitar secondhand, but this one starts off brand new.
Your sharktooth Ibby, my pointy is a monster as well, I am slowly getting to "tame" it, but I suspect the pickup swop thing is inevitable.