With the string through, just drill through on the outer holes and then drill from both sides so the holes are in a line.
Make sure a drill press has the reach to get to the hole location, small ones don’t.
At home, I find that a hard sanding block and sandpaper better than a sander for finishing headstocks. After cutting I use a bastard file to finalise shape and then paper to finish, from 80 grit to 400 finish. Keeping the edge perpendicular to the face is key.
Sounds like you are making progress!