Will dermal fillers or surgery help with my bulging eyes correction

Hi Tiffany,
Thank you for your question.  When people have deep set, hollow eyes, it is associated with too much eyelid showing.  My current preference for treating this area is Restylane.  In the past, I used fat grafting prior to 2011.  I do not like using fat for multiple reasons.  It is true that fat grafts in this area last a long time, but so does Restylane.  I have many patients who are more than 4-5 years out from a single session of upper or even lower eyelid filler with full correction.  Other patient with less than this duration may have aged or had significant weight loss or other factors which caused even more additional fat loss since the original treatment.
The main reason I prefer the fillers is maintenance.  As long as the patient is not getting injections every 3-4 months like botox or even once a year filler injections for lips and smile lines, then the duration is already very good.  
MAINTENANCE
What do I mean by maintenance?  As people live their lives and age, they may gain or lose weight.  Using the filler technique, I can inject more filler as they lose additional fat in the upper eyelid/brow area or even lower eyelid/cheek junction. I try to inject additional filler which can mix and blend together with the prior filler for a seamless appearance.  When patients get touch ups, they usually use half as much or filler as they did the first time, but not needing a full repeat of the original procedure.
WEIGHT GAIN AND LOSS
If the patient loses more weight or hollows more around the eyes, additional filler can make up the difference for the lost fat that occured over a couple or few years.  MORE IMPORTANTLY, if the patient GAINS weight, they may not need as much filler as they did before.  Their own fat cells are growing and adding back some volume.  If they look too full or puffy, using these types of reversible filler can leave room for washing out some of the filler without resorting to surgical removal of excess fat grafts.
ACCURACY
The eye area needs accuracy to the 0.1-2 cc of volume otherwise it can look too puffy or not full enough.  Fat grafts are know to have variable "take" or "survival.   Most of my younger patients may only want 1 syringe total, while other patients will naturally "puffier" eyelids have more room to hold the filler and may want closer to 2 syringes.  Some ethnicities such as East Asian patients who have naturally puffier Asian eyelids may max out around 3 syringes.  There have been exceptions to the rule, but its pretty rare.
Since you look very young, let's say you would only need 1 syringe of filler, which could be performed the same day as the consultation, with 1 day of noticeable swelling and days 2 and 3 mild swelling.  After that there is some "honeymoon" swelling which may look better than the final results.   My "process" as I perform the filling is to only fill 0.1-0.2 cc at a time and let you immediately look at the results and see the area "fill up."  The reason your input is crucial, is the only person I am try to please is you, the patient.  I may prefer a fuller look or a more hollow look, but the patient's opinion/wishes trumps my preferences.  As the filling proceeds, which is relatively painless since the Restylane is infused with numbing medicine (lidocaine) the patient begins to relax and see the process more like an art project and can see the tranformation.  I like it when the patient an I agree and see small areas of hollowing which need to be treated.  This means we are on the same page and I should be able to accurately achieved the patients desired results.
For arguments sake lets say exactly 1 cc of volume (0.5 cc per eyelid) will make you very happy.  If the survival of fat grafting is between 10-50%, then how much fat should the surgeon inject. 1 cc per eye hoping 50% survives, but what if 30% survives or only 10% survives?  If I inject 1.5 cc of fat hoping 30% survives, but only 10% survives or 50% survives (Oh no!)  The fat scars into the upper eyelid and brow area instead of being smooth and able to slide when you raise your eyebrows.  If too much fat survives, it may take a surgery to cut out the fat.  If you need more fat, the old fat is scar into place under very thin eyelid skin, so layering another bit of fat may turn out lumpy. 
Simple, Effective, Not painful, Minimal recovery, Accurate, Long Lasting, Easy to maintain, Easy to wash out (just in case).  The main negative of this procedure is that it lasts longer than I originally had thought it would last.  Instead of having the same patients from around the country/world return annually, they don't seem to come back for 2-5 years.  Of the patients who happen to be visiting New York City, I will see them for follow ups where they think they need more filler because they thought it is dissolving, but when we take after photos years later, the results don't look much different from the immediate after photos, so they usually leave without doing anything to their upper eyelid, unless they liked an even fuller look like their day after swelling.  In those cases, we simply add more filler to reach the the look they are going for.
I have a "rant" about Fat grafting below the below my signature, if you still want to read more.  But if not, I wish you the best of luck on your Facial (eyelid) cosmetic journey.  Caveat Emptor.
Best,
Dr. Yang
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JUST A LITTLE "RANT" ABOUT FAT GRAFTING.
A little known fact is that weight gain and weight loss can affect the fat grafts.  The restylane is stable in volume and ignores your weight gain or loss.  The fat grafts can grow and shrink with your weight gain or loss.  If you lost weight and the fat grafts "grew" to make up the lost volume, then that would be great!  If you gained weight and the fat grafts "shrank" to not be too full then that would also be "great", but the opposite occurs.  When you gain weight the fat grafts will also grow, and when you lose weight it will also shrink.  I am trying to be accurate to 0.1 cc, so I don't like that aspect of fat grafting at all.  Maybe in other surgeon's hands they are able to control the behaviour of the patient's fat after they leave the operating room, but I cannot, nor do I believe that I can.
I have a patient who asked me to perform liposuction for a double chin.  I listened to her.  She then added that she had body liposuction and further had that fat injected to her butt for a "Brazilian Butt Lift (BBL)" She was very happy with the results and had and hour glass figure.  She felt so good about herself that she began to workout to get toned.  But instead as she lost weight, and got skinny, her BBL also shrank away.  She paid a lot of money for the BBL, and felt it was also shrinking away, so she instead gained back the weight which put the fat back to her butt area.  But with the weight gain the fat did not return to the hips and thigh area, instead it went to the neck, shoulders and upper arms. (Look up New York Times article: With Liposuction, the Belly Finds What the Thighs Lose by GIna Kolata from 2011).  This was why she now was looking for liposuction for the double chin.  I suspect her future weight gain was no longer going to her hips and thighs, but to everywhere else on her body. 
I have tried injecting filler around fat grafts and it never looks as good as patients who never had anything done to their eyes (including upper eyelid skin removal).  The fat is hard and scarred into place and the filler runs into the fat grafts yet has trouble filling out indented areas since the fat is stuck to the sliding surfaces under the skin.
The eyelids are too delicate to mess with fat.  Large areas such as cheeks or the other "cheeks" should be fine, but when the skin is very thin, I think it is too risky.  

Patients such as yourself with good skin tone often do very well with filler along the orbital rim and tear trough.    If done well, this will blend your lid and cheek well.       It also has the benefits of lasting a long time in this area. 

I do believe eyelid filler injection will help you to reduce the bulging appearance of your eyes although there are certain limitations.