Riding Lawn Mower, Engine dies, gas in oil.

Hello, I recently purchased a 2004ish Troy Bilt Pony. The carburetor was leaking gas from the breather tube, and the oil had gas in it. So I replaced the carb and oil. Now the gas stays in the tank. However, when I start the engine, it seams to sometimes struggle, but eventually starts and runs for about a minute. Then it slowly putters out, almost like it's out of gas. If I wait about 30 seconds, it will start up and do the same thing again. I checked the fuel lines and gas cap and they seem to be fine. I also rechecked the oil and there is definitely still gas in the oil. I am not sure if it was residual from the old problem, and I am not sure if that is related to it shutting down. - Extra info, when the engine is started in choke, it puffs black smoke with each stroke. -

I am very new to working on engines and would appreciate any and all suggestions. Thank you for your help!