Welcome to our community and thank you for your question!
It's possible to apply for asylum even if you withdrew your first application. It's called a "Folgeantrag". In order to be successful with such a second application you need to provide convincing evidence to prove that your personal situation respectively the situation in Syria changed for the worse since your first application. You have to show that there is new evidence that hasn't been known or taken into account during your first process. In your case it might make sense to provide a medical report about your Schizophrenia and to tell them about your sexual orientation to show them that you would face an unbearable situation in your home country if you were sent back.
This is a complicated matter and there is a lot at stake for you. That's why I highly recommend you to get in contact with local counselling centers and/or a lawyer that can take a closer look at your case. If you tell us where you live we can help you to find such organisations.
I understand that I can also apply for permanent residency in Germany starting May 2019
I'm sorry to tell you, but I'm afraid that this won't work. The time you spent in Germany with the legal status "Duldung" does not count when it comes to an application for permanent residence permit. You need to be an acknowledged refugee or to have subsidiary protection for that.