Can't keep bike running, seems to die with throttle.

[milktatious]

Got a 2006 gz250 from a previous owner that had been sitting for a year or so. Wouldn't start or even crank, I put in a new battery, drained out old gas, cleaned the carburetor, changed the oil, and tightened the connections the exhaust pipes had to the engine. Now, the engine will fire up with only one or two starter cranks, which is good. But it'll idle extremely low until it gets to warm up, which takes a good 30 seconds or so until it idles at what I feel is a good rpm. I also have to play with the choke a little bit before it does start to idle well. After about a minute 30 seconds of idling, it'll die. The time it takes to die goes by much faster when I start cranking the throttle. After it dies, it doesn't want to start back up until I let it sit for a while.

So, I decided to get a new spark plug. The old one looked very black and sooty. The new one seemed to work well for a little bit, but I then took it back out and, after only a few minutes worth of idling and slight throttle cranks, it looks just about exactly like the old one. The image attached is the new spark plug after a couple of tries to run the engine. What I feel like is happening is either an extremely rich fuel mixture, or there is simply liquid fuel being spat at the spark plug. I think this might explain why it takes a while for it to warm up and idle properly, and why it's hard to start back up without waiting after it dies. However, I'm not sure if that's my exact issue as this is my second bike and my first time actually working on one. As such, I'm also a bit lost on how I'd go about fixing it. Any thoughts?