After finally figuring out how to apply a transparent PNG only to specific faces of a mesh, I spent forever trying to figure out why I was getting a shadow and finally realized it was because my coffee mug is hollow and a shadow is being cast on the inner wall of the mesh.
Is there a better way to add transparent graphics and logos to solid objects like this? Ultimately, I want to be able to do this for cars and characters. Thanks!
EDIT: I do not believe this is a duplicate of those other issues as my problem was a shadow casting onto the interior of a mesh due to the transparency. I tried both of the solutions shown below and here are the results. I believe the shrinkwrap mod is the best solution for me right now (thanks @masterj2001), but I will continue to tinker with the other solution (thanks @thibsert) to try to get rid of the lighter background, and I did try all the blending modes in the MixRGB node.