Isn't the whole point of moisturizer is to add oil to a face/body that doesn't have it naturally?
If you have oily skin, you don't need to moisturize. Keep your face clean and be thankful for the oil that is keeping your face naturally healthy.
I am 42 years old and have oily skin and I have NEVER moisturized - ever. I don't think I look any different/older than those who do moisturize.