Woodworking question

I'm not a woodworker. Also, I'm not particularly handy.

So. . .

I have chair. It's old. And except for a broken arm, it's in good shape. I haven't taken photos, so I'll try to describe. It looks like both ends of the arm connect with a dowel into the chair back and a post of sorts coming up from the seat. The entire arm is broken off, and the remains of the dowel are in the arm and the chair. If I can get the remaining bits of the dowel out of the various parts of the chair and arm, I can get a new dowel and reconnect and glue it into place. But the remains of the dowel are seriously ingrained/embedded in the various spots. What is the best way to remove them? Pick them out with a screwdriver, or maybe an awl. Drill a relatively narrow bit into the various part and then, again, try to pick the remains out? Saw off and then sand down the parts of the dowel sticking out and then drill into the dowel itself and then use a narrower dowel to connect the parts?

By the way when I saw dowel, I'm not sure that's precisely what it is, but that's the best word I can come up with to describe the connection between the wooden parts.

Or bring it to someone who knows what they're doing?

BTW, I looked up what the chair could be. I can't remember the style, but it's a 19th century style, and I'm guessing it's a reproduction made in the early-mid 20th century. Might be original. I don't know enough to know what to look for.

Thanks.