Firstly, are you sending these emails using third party solution? There are also two sort of bounce response, soft bounce and hard bounce. Soft bounce is when email was refused by the server and hard bounce is when the email address is invalid (non existing).
We have a field in Contact form for email activity. When you get the hard bounce, change the status of 'Email Activity' field to hard bounced and change Allow bulk email to Do not allow. At this same time, we send email to the person who added this record and let them know that email is bounced so that they can make correction.