But since that won’t happen, can anyone recomend a non-surgical treatment to remove/reduce dark bags under my eyes? What ‘product’ should I use?
I don’t think there is one. One of my daughters has this problem and was told by every expert there is that the ONLY thing you can do is to use appropriate makeup.
There have been a few threads asking this question over the years and the answer is always that there is no “cure.” Maybe some day.
Haj
You probably can’t get rid of dark circles and bags altogether via any means but cosmetic surgery but you can help them by getting enough sleep, staying well hydrated, avoiding cigarette smoking and using sun protection (both a high SPF lotion/moisturizer/eye cream and sunglasses). 
And while cosmetic products wont “cure” the problem, they can go a long way to diminishing the appearance of said bags.
Vichy has two products - Thermal S Yeux which is very good for puffyness caused by lack of sleep, alergies, crying jags, etc. and Oligo 25 Yeux which helps with darkness, over time.
If you want to refresh your eyes, so you look less sleepy during the day, either of these products is great. If you want to forever erase the bags, yer gonna have to see a surgeon.
I have this same problem. Are there any products similar to what alice mentioned that can be purchased in the U.S.?
I looked them up and Vichy is a part of the Loreal family. Just google Vichy cosmetics and you’ll find the product line.
I’m going to give that Thermal S a try. Thanks Alice.
C’mon there has to be some crazy remedy out there. A drip of pelican semen messaged onto the lids?
Well, there are always the oft-mentioned remedies (all of dubious quality, IMO) of the application of dampened tea bags, cucumber slices or cotton pads with witch hazel (aaaaaaah the burning! the burning!) which are all allegedly able to reduce the swelling and “bags” but not necessarily the dark circles.