I’m getting significanly different results when I’m choosing different methods for estimation (SANN, BFGS etc.)

This can happen. For CARMA and COGARCH the best method to use is SANN (default) because of theoretical issues about the model and the method. Anyway, increasing “trials” in advanced settings should produce similar results using different methods (see yuimaGUI user guide). Higher “trials” -> more reliable estimates.