You can get an improvement in image sharpness if you press the back of the camera firmly against your nose and cheek. The mass of your face will help damp vibrations caused by the mirror (which can show up as subtle softness even to fairly high shutter speeds). It also materially decreases camera shake to press the back of the camera firmly into your face.
Then just wipe off the LCD, if necessary.
Do you guys have fudge on your faces, or what? I have some of the greasiest skin around, and it's hardly a bother for me--when the LCD is on, it can't be seen, and when the LCD is off, it doesn't matter.