I guess another thing that can help you learning how to sing and play at once is to practice polyrhythms, or simultaneous sounding of two or more independent rhythms. You can do it just by tapping your hands over a table or your thighs. Once your brain is comfortable with that, it will be easier to dissociate your playing from your singing. I can tell music academy is not a must, but it sure helped me being a better musician. Keep practicing and you'll get where you want!