What is wrong with my controller and how do I fix it?

There seem to be a few problems with my Xbox one controller so I'll explain them separately:

I have been using a wired USB for the controller for a few months now to save on using batteries, however this week I noticed that the 'x' button has become unresponsive.

I thought that this could be due to a fault with the USB cable so I unplugged the controller and put in some fresh batteries. Once I had done this I could not get the controller to connect with the Xbox at all. The home button light would just flash twice then the controller would turn back off.

In the end I found a method to sync the controller, using the USB cable for a short period, but it only connected for a short time before saying 'batteries low' and then the controller turns off. I've tried plenty of different fully charged batteries and still get the same problem.

My controller is fully up to date and I have tried power cycling my Xbox with no luck!

Anyone know any way of fixing these problems? Thank you for your help!