I don't think you need to tell anyone outside your home about using mouthwash, especially if you are ashamed of it. Do you live with your family or roommates? If so, it may be a good idea to tell them about it because they can support you in your sobriety by making sure there is no mouthwash in the house. Also, if it has caused problems internally, you may want to see a doctor.

Best of luck with your sobriety!