Like my colleagues, I can't ethically direct you to a diagnostic instrument, when such an instrument is apt to reinforce your issues, rather than offer a solution path.
While you identify real problems in your operating culture, focusing only on the problems will not create a foundation for a healthy and thriving organization. I believe that our work in creating vibrant and life-affirming cultures should begin with discovery of the vital, essential strengths and gifts of the organization and its people. The best method for this work is appreciative inquiry, which engages the organization community with a series of exploratory questions about the instances of great success, superlative service, seamless cooperation, and more. Through this process, you are able to identify the essential positive core of the organization, the "root causes of success" behind your exceptional high points and strengths.
You can then analyze this "positive core" for the secret sauce, the ingredients that you want to draw on and leverage every single day. When these are present, your organizational dysfunction is not. You are aiming to make those high point experiences the new normal, because you are replicating the conditions and behaviors that underlie them.
The appreciative inquiry process next guides your community to imagine, to envision what is possible with this new baseline of superlative behaviors, then to name the observable, measurable behaviors that will be your daily accountability checklist of actions that you must exhibit to be in that envisioned state. Finally, you can establish a set of short-term initiatives that will speed you toward that new set of targets, knowing that your collective genius will continue to course-correct and fine tune your learning curve.
I love the appreciative inquiry process, especially because it faciliates your conscious organizational evolution toward becoming a powerfully productive AND deeply satisfying place to work.