The 20HP on a LT1000 is kind of unusual. Most of them came with a 17.5 Intek. An engine that will start/run fine for 10 minutes, but then starts to act up, is full operating temperature related. That's usually either a weak magneto, a valve adjustment, or the engine is overheating
The performance improvement with the application of choke, usually means the engine is running lean, but that doesn't always mean a carburetor problem.
Something you might want to check since it's used.... Pull the cooling shrouds and check for a mouse nest that is causing it to overheat. Take a look at your spark plug and if it looks like this, there's a good chance there's debris under the cooling shroud causing it to overheat.