I have a subtask with a due date and dependencies, but when I change the date, Flowsana does not adjust its dependent tasks' dates
This will occur if the subtask is not assigned to the project that's under Flowsana control. (By default, Asana doesn't assign subtasks to the project of their parent task, so you have to manually perform the project assignment.)
The way Asana's programming interface works is project-based; that is, an external application is notified of changes to tasks in projects that the application asks for. When you change a subtask that's not assigned to the parent project, since the subtask is not in the project, Asana never notifies Flowsana that the subtask has changed.
So, if you want to have subtasks participate in Flowsana's automations, you'll need to assign the subtasks to the project that's under Flowsana control.