So the problem is too much fuel. Assuming that your air filter isn't plugged and the air box lid doesn't have any restrictions, I am betting that you can fix the problem with a carb rebuild. Good chance that the needle valve is leaking so it is always trying to empty the fuel tank into the carb. Makes it run really rich so with the air filter and air box together it can't get enough air.
Mine: 2018 Sportsman 850SP - 26" Bear Claw HTRs on OEM steel wheels(I really don't like the Aluminum ones)
Polaris Hand warmers, Mirrors.
Hers: 2007 Sportsman 500 EFI, Polaris Winch, 26" Bear Claw HTRs on Moose 427x Wheels. QSC Clutch Kit, Hand Warmers
2001 Magnum 325 MOSE 12" rear wheel conversion. 25" Sedona Coyote tires
2001 Magnum 325 Warn winch and Moose 60" plow
2017 Polaris Ranger 1000XP - Roof and front glass
SOLD 2004 Sportsman 600 Twin