First-time painter as well so I would defer to the pros, but I would rinse the gun between coats of the epoxy every time. I wouldn't disassemble, but I rinsed it out with lacquer thinner and blew that through a few times with air. I also was using the DeKups system towards the end, so that allowed me to take the cup off instead of dumping the epoxy back into the mixing cup, but I would also dump the epoxy back into the cup.
I was just too afraid that the paint would gum up in the flow path and I would fling out partially dry epoxy spots. I suspect that's what you are seeing, but again, I'm no pro.