Ok, I'll try that out and see if it works. Thanks for the advice.

EDIT: Ok, tried that out by placing the layer with my drawing on it (just a few quick paint lines for testing) above the layer with the UVs on it (I changed the UV section from a background to a layer) and turned the UV layer off or invisible by clicking the eye. Now, when I apply it to the giraffe, instead of showing the rest of the UVs with the paint on them, I only see the paint (but still in the small size that I saw it as before).