Hi, welcome to the Guild!

I'm guessing the problem lies with the combined paths. Although a combined path will act as one object, it isn't truly one single, continuous path. From looking at your pic, you have open paths that either start or end within the main shape. Inkscape doesn't know how to break apart your shape when the open paths don't extend beyond the edges.

The Division tool should definitely do what you want however, but I'd suggest trying to break apart each continent one at a time, rather than trying to do them all at once. You could perhaps start with the area towards the bottom left, where it looks like you have a closed path, and take it from there. If you're not using a closed path, try to extend your lines beyond the edge of the main shape to avoid problems.

Hope this helps!