Exactly the same thing happened to my friend, it's infuriating, bureaucratic crap. What about alternating weeks? Or seeing the other one for blocks of 5 across the year. FWIW, my friend found the time went fast and that she gained something from seeing a different T for a bit (even though the new T was a nutjob and she was glad to get back to her old T).
It sucks, really sucks and I'm sorry you have to go through it all. I hope you manage it in a way that works for you.