I think the problem lies with the amp you have. In my own experience, the Marshall non-tube (Solid State) amps sound terrible, overly saturated and no clarity whatsoever, just very muddy sounding. Fender non-tube (solid state) amps are really good sounding, almost like tube amps. I know you may have been influenced because Billie Joe uses Marshall, but I've been down that same road back in the 1990s. I bought Marshalls purely because all of my favorite guitarists used them, but I hated the sound that came out of those amps. Soon as I played a Fender amp, I finally got closer to the sound I was searching for.
I'd recommend trying out some Fender amps, they are my amp of choice and can get better Green Day tones, especially Know Your Enemy. Stay far away from distortion pedals, they are crap and will ruin your sound. Just my 20 years of guitar playing experience.
But if you still get can't the tone, you are not alone. Most guitarists can't nail the tone of any of their guitar-gods. Eddie Van Halen once said that he was a huge Eric Clapton fan, and when he was growing up, he would play Eric Clapton songs "note for note" and never could get it to sound right.