After seeing your photo, I see very beautiful eyes but it is apparent to me that your eyelids aren't heavy, as you suggested, but rather they are hollow and deep-set. The hollowness is more pronounced in your right eye. Injecting a little bit of Restylane in the upper eyelids will soften your look, even out the asymmetry and restore a more youthful appearance. This treatment should not be performed by just anyone. Injecting around the eye and especially in the eyelid area is a delicate and highly-skilled procedure and is something that I do in my practice on a regular basis.
Finally, I strongly suggest that you not let anyone cut on your upper eyelids. There is nothing to take out and you will only be left with more hollowness.