How can I get rid of my under eye fat pad/puffiness? It's like there are two sets of bags.

Eyelid puffiness. Depending on the appearance of the eyelids, there are different treatment options. If the puffiness is from fluid retention, a low salt diet may help. If it is from fat protrusion, this can be camouflaged with filler or can be repositioned with a lower eyelid lift.
Non-surgical ways. Dermal fillers such as juvederm ultra xc are an excellent alternative surgery for treating the bags underneath the eyes, or "tear trough deformities". They restore volume and give you a natural look which can last for up to one year.