public string GetConnectionString()

{
    //This is the existing layer for which we want to switch the underlying datasource
    var lyr = MapView.Active.Map.GetLayersAsFlattenedList().OfType<FeatureLayer>().FirstOrDefault();
    //Note: needs to be called on the MCT
    var connectionStringToReplace = lyr.GetFeatureClass().GetDatastore().GetConnectionString();
    string databaseConnectionPath = @"Path to the .sde connection file to replace with";
    //If the new SDE connection did not have a dataset with the same name as in the feature layer,
    //pass false for the validate parameter of the FindAndReplaceWorkspacePath method to achieve this. 
    //If validate is true and the SDE did not have a dataset with the same name, 
    //FindAndReplaceWorkspacePath will return failure
    lyr.FindAndReplaceWorkspacePath(connectionStringToReplace, databaseConnectionPath, true);
}