I did a little bit of research about Dublin III. This is what I found out:
When you apply for asylum in a country that's part of Dublin III, it will check whether it's responsible for your case or not. If you leave the country demonstrably for more than 3 months and apply for asylum somewhere else before they finish checking, the country where you applied for asylum a second time is in charge of checking which country is responsible for your case. It doesn’t automatically mean that they will process your asylum procedure - they will only do so if they come to the conclusion that they are responsible.
Since you’ve already been rejected by Dutch authorities based on Dublin III, it has already been worked out which country is responsible to give you a procedure.