Do I have exophthalmos? (Photo)

See an an orbital oculoplastic specialist for evaluation. The most common cause of bulging eyes is thyroid disease followed by genetic (which can look worse with age as the fat around the eyes goes away). See following video.

Hi and thanks for the photos and question
You have proptosis which is protrusion of the eyeballs from the eye sockets.  There are many potential causes of proptosis including Graves disease which causes thyroid problems and deposition of immune complexes in the soft tissues of the eye socket behind the eyeball and proptosis (exopthalmos only if you have Graves disease).  Other causes of proptosis include tumours in the brain or meninges (Meningiomas) nerve sheath tumours and neurofibromas.  It sounds like you ned to have some hormone tests and scans of your orbits, brain, pituitary gland and abdomen (Adrenals).
Good luck
Dr Mark Hanikeri MBBS, FRACS

Be sure to have your thyroid function tested as this is the most common cause of exophthalmos. Lid retractor recession is an option that would allow the lids to relax and close better. This type of surgery may be done with local anesthetic and sedation. Orbital decompression, removal of orbital bones to allow the eyes to recede back into the sockets would require general anesthesia and involves more risk. If your vision remains good, the eyelid surgery might be a better option for you. Best wishes.