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();
QueuedTask.Run(() =>
{
  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);
});