Are eye bags caused by sagging skin and fat bags?

This is a good question. Cryolipolysis (Coolsculpting) can freeze the fat around your body and make it shrink, including in some cases around the face. The fat around the eyes is not a good area to have this treatment because the freezing technique is too close to the eyes and, even if it was safe, it would be difficult to calibrate the amount of fat you are trying to make go away. Your best bet would be to get an opinion from a qualified oculofacial plastic surgeon who is part of ASOPRS (American Society of Ophthalmic Plastic and Reconstructive Surgery) who focuses only on eyelid surgery to determine what treatment would help the best, be it laser, filler, or surgery. There are many great specialists in your area.

Tear troughs, depressions across the lower orbit with bulging fat pushing out the skin forming what many call "eye bags" are caused by loss of volume in the cheek and stretch of the skin of the eyelid and cheek. as the skin descends, it is stopped by ligaments from the orbital rim to the skin. The bulging is caused because the skin of the eyelid cannot descend any more and the normal fat of the orbit bulges into it and because the cheek skin is suspended by the ligament and the residual fat descends to the inferior ligament(s). Freezing, or otherwise destroying or removing the fat, only makes the problem of volume loss worse. Although you will look better immediately, you will eventually look much worse and need an increased amount of fill. Surgeons have also tried skin tightening techniques for years, but none other than surgical removal of the excess skin have been successful for longer than a few months. Instead of wasting your money on things that don't work, see a plastic surgeon who is an expert on fillers (including fat) and surgical blepharoplasty and discuss options that really work.