She’s embarrassed to be seen with me in public
You have a valid point and I do think it is hypocritical. Have you tried asking her this question? If she can’t be with you in public, why should you continue with this? Adultery can tear your relationship apart, and you really do owe it to her to come clean if the two of you have been honest with each other until now. How she will react depends entirely on the kind of relationship you have with her. You made a mistake and acknowledge that. You also recognise that not telling her about it is making it hard for you to function. I can’t tell you to lie, because that is never a good idea. You have to decide whether you want to be open about this or withhold information and live with the guilt of that decision.