Is Botox Really Meant for You to Look Younger?

Botox, when properly injected, does create a more youthful appearance. It can elevate sagging brows, lift the outer edges of the eyelids, reduce the severity of crow's feet, etc. However, if you have deep wrinkles or bags under the eyes, the treatment of choice would be Restylane. I don't know where you were treated, but any board certified dermatologist or plastic surgeon would have a variety of fillers or lasers in his arsenal.

Botox is injected into the muscle.  The muscle stops moving so much, and the wrinkle on top of it usually smooths out a bit.  Like any treatment, some will get a great result, and some, not so much.  It sounds like you have extra skin under the eye.  Botox is not a great tool for this.  It is a usually good tool for the area between the eyebrows.  Best to see a couple of board certified surgeons who operate on the lower lids and see what they think.

Botox is intended to relax the muscles that contribute to the formation of wrinkles. And this can help individuals look younger (see video).

Botox is really not recommended for treating under eye wrinkles, which primarily develop from sun damage as opposed to overly active muscles.

However there are other options that can help counteract the damage caused by UV rays. This is done by helping the skin to develop new forms of collagen. Fraxel laser systems can help in this regard. Also you may want to consider Radiesse as an undereye injection. It works as a wrinkle filler. But it also helps promote the development of new collagen. 

Ask your dermatologist about these options.