Runs perfect for the first 10 mins then dies a terrible death

Hello all,

My question is simple really. I have a 98 Sportsman 500 that is a real chick magnet. The problem, when I start it up it runs perfect for about 10 mins then it chokes and coughs and sounds like its dying. Then it will back fire and make terrrible noises.

If I turn it off and let it get cold it does it all over again.
Some threads on here address the issue but none have the solution.

Interestingly enough I did recently adjust the transmission key tab because she wasn’t going into gear properly due to a crack. I bought a new one and installed it and it ran flawlessly. Then this started. I have heard that its caused by the tranny thinking its in reverse. I have also heard its the dirty carb. I have purchase a new carb kit but have not had the chance to install it.

Any thoughts? The correct answer will win you a naked pic of my dog!