Hi VJ -  Have you talked to your dermatologist about trying spironolactone (decreases testosterone slightly in women but still keeps it in the normal range)?  It will decrease your oil some. Your hormones can be tested first (a blood draw) and it may be helpful depending on what your tests show.  Also, those inexpensive salicylic acid acne washes (usually 2%) in the drugstore really do help dry up oil a little.  And, the benzoyl peroxide gels (also drugstore) can be drying also.  Make sure all your products are oil-free and if you use a moisturizer it's a gel.  You might like this hyaluronic acid based one - a lot of our oily patients love it - hydration with no oil. 

Also, if you've had bags under your eyes since birth (really?? :)), I think you will probably need plastic surgery for this.  Find the best eyelid surgeon in your area and get a consultation - you may be happier with this route. See my article on eye issues for more information.  Hope this helps,  Dr. I