Embroider around the painted design. Some paints dry soft and flexible. Acrylic paint, however, dries hard. If you are stitching through the painted area, bring the needle through with caution, as making excess holes will show in the paint.
It took Michelangelo 4 years to finish the Sistine Chapel so give yourself at least a good couple of years of practice to produce your first coloring masterpiece! You need to invest in your coloring hobby to reap the rewards! I don't mean you need to go spend your life savings on the most expensive art supplies or furnish part of your house with an art studio, I mean spend the time to allow yourself to become better at coloring. Most things in life take practice and coloring is no exception.