So I wish I was giving better news, but that unfortunately isn’t the case. The good news that the face is fully rerigged, but this is quickly overshadowed as you can tell from the screenshots above.
So whats happening is the textures of the eyes are staying in they’re default location and doesn’t follow the meshes they’re assigned to at all (both movement and rotation), and gives an overall unappealing effect (basically the eyes broke in a way I didn’t even know was possible). The most puzzling thing about this is the mouth is fine and acts exactly like you would expect it to!
I have looked through the textures to make sure there was no obscure setting was enabled in the eyes and not the mouth; and as far as I can tell there is nothing different between the eyes and the mouth. This has left me with a singular conclusion: Just liked how the facial rig broke in the first place (which I mis-diagnosed the cause originally), Blender changed how the “Mapping” node works in 2.80 to 2.81, and while visually it looks very similar to before, I have a feeling they changed something ‘under the hood’ that is causing the eyes to freak out in Eevee… This doesn’t explain why the mouth is fine, but it’s the only logical answer I have on my current situation.