Why do my tears burn/sting on the skin around my eyes? the direct contact hurts and the area hurts the next day. this is new to me.

Dermatitis: You could have a dermatitis around the lid margin. This should be evaluated and the appropriate treatment instituted to resolve this.

Chapped skin: Most likely you are developing a dermatitis due to the constant moisture. I have had success with many patients by having them wash and dry the area twice a day, followed by placing a thin amount of petroleum jelly (vaseline) to the affected areas. Once it clears you can discontinue after 5-7 days.