None of our users were actually dropped out of their sessions (that we know of). We were getting the same error messages, it was taking up to 5 mins for a connection server to hand off a request and the connection servers were not able to power on machines if they were off. All very weird.
In the end we powered off all of the connection servers and powered them up 1 at a time. Waited until all services were up and functioning before powering on the next. Magically, it's all working again.