I assume you've pulled out the exhaust screen Next thing. Have you pulled the cylinder and looked at the piston closely? I had a 362 come in the other day that had a chunk broke out of the piston directly in line with the intake port at the base of the piston. I pulled the exhaust and there was no scoring, pulled the intake and there it was. We could actually get the saw to run at times, then it would mysteriously die. All the pieces hammered one of the transfers and puked out the exhaust, so there wasn't anything rattling around. It honestly pounded or burned the pieces up small enough to go through the screen.
My next thing would be to find a carb system I could switch out and try.
My next thing would be to find a carb system I could switch out and try.