Although project management training is great and I would recommend Agile or Scrum methodologies to assist you with your challenges, it sounds like the issues you are having may be related to needing some definition around roles and responsibilities, setting expectations and communication of the impacts of a slow response time. Using different methodologies can definitely help with that but in the near term maybe you just need some coaching in the softer skills to get through your immediate challenges.

I am a project management coach and would be more than happy to give you some pointers to help you out.