updated ipod touch 4g from 5.1.1 to 6.1.6 - now imessage and facetime barely work

I recently updated my iPod Touch to 6.1.6. During the update, I was unable to properly set up the Apple ID for FaceTime/iMessage (later I learned this was because it had apparently brought up a similar but different email address), but continued the update and went back to Settings to sign into my current Apple ID. Before the update, I had no problems with either program, but now iMessage refuses to send messages on my home network most of the time, in areas where I still had a connection to the router and could previously use them. The same goes for FaceTime.

This problem has extended to a few away-from-home networks, and usually resolves itself, though I am never able to replicate the actions that allow it to function properly.

I've attempted a wipe and restore, have cycled the power to the iPod, reset the router for the home network and signed in and out of my Apple ID account across the board, and the problem persists. I'm feeling like I'm running out of options but am not sure if I've missed anything. Should I be contacting support/looking for a new device? Or is there something I can do to alleviate this?