Can't activate phone, cell connection not working.

My iPhone 5S recently had a bad accident where the display cracked and the entire chassis got bent slightly. I bought a replacement display, attached it and the phone was useable again. The camera and the loudspeaker were dead and the multi-touch was acting up intermittently, but I could use it.

After a while I started getting problems with the cellular connection. All of a sudden it got stuck on "Searching...". A reboot took care of it.

Then all of sudden, while being used, the phone reverted to an "unactivated" state and required me to activate it before I could do anything else. The cell wasn't working but when I connected to a Wifi I could activate it and everything went back to normal.

The next day the same thing happened again and now I'm unable to activate it even on Wifi. I've tried several different Wifis and it's always the same, it tries to activate for a couple of minutes and then I get an error saying Apple's servers could not be reached.

When I connect the phone to iTunes I get an error message and "Please try again later"

The phone is obviously a lost cause and I have most of the stuff on it synced to other devices, but the last backup is old and I have some voice memos on it that I'd like to get hold of.

Does anyone know any way to get those from the device? Is it possible to make a backup of the phone in this state?

Thanks!