Droopy eyelid, and my eyes feel like they've been pushed in. What procedures/tests do I need to fix the eyes? (Photo)

I have been doing all kinds of eyelid surgery for 34 years. My strong recommendation it that you do not do anything. Your eyes look very nice. With extremely minor imperfections the margin for success is very narrow and it is very likely that things can go wrong.

It is hard to tell from the 2D photos but it appears your left eye is more sunken than the right eye, which was likely present prior to the massage you had. There is also mild left upper eyelid ptosis (droopy) which may be worse. Treatment may or may not be necessary. Best to seek in person consultation by an oculoplastic specialist.

I agree with your consultation that you need to wait a period of 6 months to see if your eyes will recover from the massage.  If the eyelid hooding persists a blepharoplasty would be the proper procedure to remove the excess skin.  If you have eyelid ptosis that may be repaired at the same time.  Please consult a plastic surgeon who is experienced an oculoplastic surgery.