Ok, from what you said, I can see that you have to use anesthesia sometimes and you are worried about the potential threats to your eyes. Anyway, it is true that anesthesia would give rise to some eye problems because it would inevitablty affect your visual nerves or your neurological system, which leads to some blurry vision, watery eyes, etc. But slight amount of it would be fine, just try to protect your eyes and be careful of what you take in.
I never heard that anesthesia can cause eye problems. So, i just searched it online and found that some people reported double vision or other vision problems by using of anesthesia. But still some doctors and anesthesiologists said they have no connection. Anyway, there are some people experience vision problems because of using anesthesia. So, i can't tell if using anesthesia cause eye problems or not. If you need to use it, you'd better consult your doctor to make sure. I believe s/he know better than us.
Yes, using anesthesia will lead to some eye problems. However many of them are just temporary which will be reduced after a while. You don't have to worry. If your skin is so sensitive, you may have the eye problems, like the dryness of the eyes after using anesthesia. You may have the swollen eyes which could be cured with warm compress. You just wait for a while and protect the eyes carefully.