Seems like you may have hit a similar to issue to this one: App not responsive, website down, cannot deploy!
Is there any way to either manually manage these apps more directly
You can switch to using Fly Apps v2 aka Fly Machines (what you’re currently using is Fly Apps v1) if you desire more control: Difference between Apps and Machines - #4 by ignoramous
…or a more idiomatic way to get back into a normal state?
I believe something is up with the Fly control plane… I suggested this in the other thread (linked to from above): Deploy the same code to a new Fly app (v2 instead of v1, if you are feeling adventurous).