I do not know if this will help, but I think what you are experiencing right now is similar to what is called Body Dysmorphic Disorder (BDD). I am a psychology major, but aside from what little I know, I cannot tell you whether you have that disorder or not, except that what you say is very similar to it.
I really think seeing a psychotherapist will help you. I know it feels very hard to hurt and disappoint your family, but if this obsession gets to the point where maybe you can't even talk properly to people anymore... Then, if this gets worse, it will not only strip your life of any normalcy, it will also affect your family and make it difficult. So in the end, they will get affected whether you deal with this now or later.
I'm glad that you know your thoughts are not rational and logical. I am also sure that you have very pretty eyes and they are evenly matched, or else you would have already been picked on and laughed at long before as a kid. No matter what happens, try not to believe the thoughts coming into your head, because to let yourself believe it may lead to self-harming behaviors (such as wanting plastic surgery, hurting your eyes).
I know psychologists have therapies to help those with obsessive complusive disoder, and this is similar to it, only what you are obsessing over are your eyes, and you are being compelled to do certain actions because of it (keep looking until it hurts). The best way would be to go see a psychologist, but you can try stopping the compulsive actions such as looking in the mirror or avoiding gazes. They will be hard to do, though, so if you are struggling to stop your compulsive behavior, I suggest you get some professional help. Hope this helps.
I really don't think going to a therapist would have as bad an effect on your familly as you think. You are keeping all these thoughts and feelings in your head and not expressing them, thus they become amplified and distorted, blown out of proportion. Loving, caring family members would most likely think 'I hope she gets better and if seeing a therapist helps that, then good for her' rather than wanting you to not see one and continue to suffer. You could also go see your regular clinician by yourself to discuss this matter and they could advise you, you don't have to go straight into seeing a therapist.