I think what has happened is that in the process of converting your image to a vector not all areas were filled in with white. You can check that by changing the Stage color in the Properties panel to something like gray so you can see. The next issue is that the shapes where you want to apply color are not closed. If you're using the Paint Bucket tool, you can set your options lower in the Tool panel to whichever applies:
- Don't close gaps
- Close small gaps
- Close medium gaps
- Close large gaps
Then you can use the Eraser tool to remove any of the undesired overfill. You can set options for the Eraser, too, so that you can:
- Erase Normal
- Erase Fills
- Erase Lines
- Erase Selected Fills
- Erase Inside
You might also use the Brush Tool to apply your fills with similar options:
- Paint Normal
- Paint Fills
- Paint Lines
- Paint Selected Fills
- Paint Inside
Hope that helps!