I would be tempted to try and use a flexible drill bit. I have a small cheap set that are about 6" long. Can't find a picture of them online.
But there are many different kinds out there and many different price points but here's an example of them.
Either there was an edit or I missed the part about filling and redrilling the old holes.
My two recommendations are either find dowels the correct size and glue them in, and sand them flush, or get some wood filler (preferably stuff made with real wood) and fill them up well, using a nail to push it all the way in. Let it dry and then sand it flush.
if the holes are bigger than 1/4" I'd tend to want to go with the dowel approach.