The quickest way to inadvertently break your eSync relationship is by altering your sending or receiving web site's connection string. Why? Because both strings are written to the database at the time the sync relationship is created. When you sync, the Ektron Windows Service grabs this information and tries to verify it. When the service finds that the new connection info does not match an existing one, sync fails. Depending on your version, you might see errors ranging from "Error reading/validating eSync license key", "Faulted State" errors, "no site details found" or even version mismatch messages.
So what can you do to avoid this? Two things; the first of which is modifying the existing relationship and profiles found in the database.
If you've never gone searching for where your sync settings are stored, let me help; go straight to the scheduler table. Every relationship or profile you're ever created will be stored here.
Start by backing up the database you will be adjusting and all databases for sites directly or indirectly that the site would be syncing with. Now, make your way to the most important column, triggerAction. This column holds all the necessary information about your sync relationship or profile including connection strings, conflict policy, and direction. You're going to need to match the alterations from your web.config's connection string to the triggerAction. The connection string information can be found within the <LocalSite> and <RemoteSite> tags. My suggestion is to open the table in "edit" mode and copy the column to notepad. Make the necessary fixes and then paste the xml back into the column within the database. Make sure you fix the relationship row as well as the existing profiles, else profiles you create in the future will still contain the old connection string data.