Yep. You shouldn't be able to pull the governor shaft out. I can't be sure exactly what's broken or stripped out, but something is. If you remove the engine from the generator you can fairly easily get to the governor to see the problem.
If you'd like to try to get to the governor assembly, let me know the engine model, type and code, and the generator make and model, and I'll explain how to get to it.
It sounds as if the governor shaft bushing is bad also. This is probably why it was leaking. It may be that you can push the governor shaft back into position and have the engine run fine. But I wouldn't try it. The shaft will probably dislodge again, while running, and you'd have no more engine. If there's something broken inside, then it will likely come apart, and again, no more engine.