Moisturizer should not leave your face oily - if it does, you're using the wrong product. Aim for one that leaves you with a healthy, hydrated feeling.
Typical moisturizers come in lotions, creams, and ointments. Creams and ointments can be oil-based. This is great for dry skin but the worst option if your skin is already oily.
There are options besides the traditional lotions. Alternative choices can leave your skin moist without the same heavy, smothering sensation that lotion sometimes brings. Oils such as almond, coconut, or jojoba are common choices. For oily skin, aloe vera gel is an ideal choice. It absorbs quickly into the skin, sealing in moisture.
The only way to actually fix the problem is by regular exfoliation. When you reveal the healthy, glowing skin underneath, you may realize that you never needed the moisturizer at all. Just remember that over-exfoliation can do more harm than good. Start only using a few times a week, building up to more frequent use only if needed.
Hyaluronic acid is a great serum for oily skin. It hydrates but also leaves less residue behind than other serums. This keeps your pores from clogging or leaving your skin with a dull finish. This blend adds vitamin c into the mix, which brightens the skin.