Which procedure would help correct my asymmetrical eyes?

Generally it is impossible to cosmetically adjust the relative heights of the two globes nor is it even necessary. This type of asymmetry is very common and generally it does not bother most people nor noticed by others. While there are surgeries like floor implants or even more radical surgeries that could adjust the relative heights of the eyes, the complications associated with these procedures are too common and severe to warrant these interventions except in cases where the physicial discrepancy is dramatic. This difference is not dramatic. You do have upper eyelid ptosis and compensatory eyebrow elevation. Surgical intervention would make your eyes brighter. However what you are describing that the asymmetric in the position of the globes prevents you from socializing puts you into some of the criteria for body dysmorphic disorder. Depending on the severity of these thoughts, the assistance of a mental health professional is warranted.

It is difficult to detect asymmetry with the photo given. A facial plastic surgeon's office can take standardized photos to assess the face in a systematic fashion (controlled lighting, facial position, background). That said, everyone has asymmetry of the bones of the face. The eye and the tissues supporting it are important to vision. This is why any procedures around the eye including under eye and tear trough filler are done with special techniques to minimize affecting vision. It is rare to operate on the eye sockets unless there was a traumatic injury. In some cases, the eyes can bulge from sinus (especially in the Southeast US) or thyroid medical problems.

All people have facial asymmetry. There is no person who has a symmetrical face. For most people, the left eye socket, consistent higher in the skull than the right.
During embryological development, the two sides of the faith develop independently from each other, and eventually fused in the midline. The human brain is accustomed to seeing human faces as being asymmetrical, since no one has a symmetrical face.
If you look at computer generated symmetrical faces, you may notice they have an odd unfamiliar appearance because they don’t exist in real life.
Your face is normal and very much within one standard deviation.
This should not, nor does it need to be treated.
Best,
Mats Hagstrom, MD