My mom always find dirty dishes an eyesore and wants to wash them right away. I have an eyesore (I have sore eyes but only one eye sore) You eyesore correctly in the first sentence but not the second. Eyesore only has one definition, which is when something is so ugly that you cannot look at it. It makes your eyes sore. For the second sentence, you can say, "My eye feels sore." Since it's only one eye, you use the singular form. You can also say, "My right/left eye feels sore," if you want to be specific.

My mom always finds dirty dishes an eyesore and wants to wash them right away. I have an eyesore (I have sore eyes but only one eye sore).