Allot of things can cause that feeling. I've felt old since I was nine. Although I do have an old soul, the feeling was actually caused by stress.
Souls don't really age the way humans do, they simply become more knowledgeable and experienced. We call souls "old" when they have been through so many lives, but really a soul could be created fifteen years ago and(if misfortune is with it) go through more lives then a soul that's over fifteen billion years old. That's a pretty wide gap and perhaps unlikely, but the point is the same. My point is that an "old soul" wouldn't result in you feeling old because the soul does not get old in the same way humans do.